Sharp Critical Alexandrov Stability and Short-Time Heat Flux

1. Introduction

Alexandrov’s Soap Bubble Theorem states that a compact embedded hypersurface of constant mean curvature in Euclidean space is a sphere [1]. A central quantitative question is how closeness of the mean curvature to a constant controls the deviation of the hypersurface from a sphere. For a bounded domain $\Omega \subset {\mathbb{R}}^N$, write $\Gamma =\partial \Omega$, let H denote the normalized mean curvature, so that $H=1/R$ on a sphere of radius R, and set

$$H_0={\displaystyle \frac{\left|\Gamma \right|}{N\left|\Omega \right|}}.$$

For $z\in \Omega$, define

$${\rho }_e\left(z\right)=\underset{x\in \Gamma }{max}|x-z|,{\rho }_i\left(z\right)=\underset{x\in \Gamma }{min}|x-z|.$$

Thus ${\rho }_e\left(z\right)-{\rho }_i\left(z\right)$ is the thickness of the smallest concentric annulus centered at z that contains $\Gamma$.

Recent work of Gonçalves da Silva and Poggesi [5] gives a complete $L^r$ quantitative Alexandrov profile on $C^{k,\alpha }$ classes. The behavior changes at the critical exponent

$$r_c={\displaystyle \frac{N-1}{2}}.$$

For $r>r_c$ the estimate is linear, while for $r<r_c$ it is governed by a sharp power. At $r=r_c$, their estimate contains a full logarithmic correction. More precisely, in the critical case their bound has the form

$${\rho }_e-{\rho }_i\lesssim {\parallel H-H_0\parallel }_{L^{r_c}}max\left\{log{\displaystyle \frac{1}{\parallel H-H_0{\parallel }_{L^{r_c}}}},1\right\},$$

up to the regularity-dependent normalization appearing in their theorem.

Our first contribution is a refinement of this critical regime. The key observation is that the nearly spherical reduction in [5] already yields critical control at the $W^{2,(N-1)/2}$ level. Instead of passing through $W^{1,N-1}$, we apply a Brézis–Gallouet–Wainger inequality directly at the critical $W^{2,(N-1)/2}$ scale; see [2,3]. This lowers the logarithmic loss from the full power 1 to the fractional exponent

$${\gamma }_N={\displaystyle \frac{N-3}{N-1}}.$$

Theorem 1.1 

(Refined critical Alexandrov estimate). Let $N\ge 4$, $0<\alpha \le 1$, and let Ω range in a fixed bounded uniformly $C^{2,\alpha }$ geometric class. Set

$$p={\displaystyle \frac{N-1}{2}},{\epsilon }_H={\parallel H-H_0\parallel }_{L^p(\partial \Omega )}.$$

There exist class-dependent constants $C,C_{*}>1$ and a point $z\in \Omega$ such that

$${\rho }_e\left(z\right)-{\rho }_i\left(z\right)\le C{\epsilon }_H{\left[1+{log}^{+}(C_{*}/{\epsilon }_H)\right]}^{{\displaystyle \frac{N-3}{N-1}}}.$$

(1)

The logarithmic exponent $(N-3)/(N-1)$ is optimal within uniformly $C^{2,\alpha }$ families.

The upper bound follows by combining the $W^{2,p}$ estimate in the critical nearly spherical reduction of [5] with the critical Brézis–Wainger exponent $(p-1)/p$. Optimality is proved by a new family of logarithmically concentrated radial perturbations of the sphere. For that family, the annular defect and the critical curvature defect satisfy matching asymptotic relations, which force the power $(N-3)/(N-1)$ in (1). Thus the improvement concerns the critical asymptotic profile itself, not merely its multiplicative constant.

Our second contribution is a parabolic application in which direct curvature measurements are replaced by short-time boundary heat-flux data. Consider

$$\left\{\begin{array}{cc}{\partial }_{t}u=\Delta u,\hfill & \mathrm{in}\Omega \times (0,\infty ),\hfill \\ u=0,\hfill & \mathrm{on}\partial \Omega \times (0,\infty ),\hfill \\ u(·,0)=1,\hfill & \mathrm{in}\Omega .\hfill \end{array}\right.$$

(2)

The boundary heat flux

$$q_t={\partial }_{\nu }u(·,t)$$

contains local geometric information at short times. Constant-flow problems of this type are related to heat content and isoparametric geometry; see Savo [6,7,8]. Cavallina and Pinamonti [4] studied a discrete-time overdetermined version in which the flux is constant only along a sequence of times. For sequences accumulating at zero, short-time heat asymptotics detect the boundary geometry. In related work, Rafik Zeraoulia [10], in a paper accepted for publication in the Journal of Mathematical Analysis and Applications, extended the qualitative short-time rigidity theorem to $C^2$ boundaries and established the finite-regularity pointwise expansion needed for that argument: if

$$H_{\Sigma }={\kappa }_1+\dots +{\kappa }_{N-1}$$

is the sum of the principal curvatures in the inward-normal convention, then on a bounded $C^2$ domain

$${\partial }_{\nu }u(y,t)=-{\displaystyle \frac{1}{\sqrt{\pi t}}}+{\displaystyle \frac{1}{2}}{H}_{\Sigma }\left(y\right)+o\left(1\right)$$

(3)

uniformly on $\partial \Omega$ as $t\downarrow 0$. Under $C^{2,\alpha }$ regularity the remainder is $O\left(t^{\alpha /2}\right)$. Zeraoulia’s accepted paper is qualitative in nature: it establishes rigidity from exact short-time overdetermination. The present manuscript is independent work by the present author and does not revisit that theorem. Building on the finite-regularity expansion, we establish a version uniform over bounded geometric classes and use it quantitatively, in combination with the sharp critical Alexandrov estimate proved above.

Define the distance of the observed heat flux from the constants by

$${\delta }_r(t;\Omega )=\underset{b\in \mathbb{R}}{inf}{\parallel {\partial }_{\nu }u(·,t)-b\parallel }_{L^r(\partial \Omega )}.$$

(4)

Writing $H=H_{\Sigma }/(N-1)$, we prove

$${∥{\partial }_{\nu }u(·,t)+{\displaystyle \frac{1}{\sqrt{\pi t}}}-{\displaystyle \frac{N-1}{2}}H∥}_{L^{\infty }(\partial \Omega )}\le C{t}^{\alpha /2}$$

(5)

uniformly on the class. A Minkowski-normalization argument then converts the best arbitrary constant in (4) into the geometrically distinguished value $H_0$, giving the transfer estimate

$$\parallel H-H_0{\parallel }_{L^r(\partial \Omega )}\le C\left({\delta }_r(t;\Omega )+t^{\alpha /2}\right).$$

(6)

Combining (6) with the quantitative Alexandrov theory of [5], and with Theorem 1.1 at the critical exponent, gives the following complete heat-flux profile.

Theorem 1.2 

(Short-time heat-flux stability). Let $N\ge 2$, $0<\alpha \le 1$, and let Ω range in a fixed bounded uniformly $C^{2,\alpha }$ geometric class. Suppose

$$r>1,r\ge {\displaystyle \frac{2N-2}{N+1}}.$$

Set

$${\eta }_r\left(t\right)={\delta }_r(t;\Omega )+t^{\alpha /2}.$$

Then, for all sufficiently small $t>0$, there exists $z=z(\Omega ,t)\in \Omega$ such that

$${\rho }_e\left(z\right)-{\rho }_i\left(z\right)\le C{\Phi }_{N,r,\alpha }\left({\eta }_r\left(t\right)\right),$$

(7)

where

$${\Phi }_{N,r,\alpha }\left(s\right)=\left\{\begin{array}{cc}s,\hfill & r>{\displaystyle \frac{N-1}{2}},\hfill \\ s{\left[1+{log}^{+}(C_{*}/s)\right]}^{{\displaystyle \frac{N-3}{N-1}}},\hfill & r={\displaystyle \frac{N-1}{2}},\hfill \\ s^{{\tau }_{N,r,\alpha }},\hfill & r<{\displaystyle \frac{N-1}{2}},\hfill \end{array}\right.$$

for a class-dependent constant $C_{*}>1$, and

$${\tau }_{N,r,\alpha }={\displaystyle \frac{2+\alpha }{2+\alpha +{\displaystyle \frac{N-1-2r}{r}}}}.$$

(8)

All three profiles are sharp in the corresponding regimes.

The subcritical power and supercritical linear profiles are inherited geometrically from [5]; we show that their optimality persists after replacing direct curvature data by the parabolic observable (4). In the critical regime, both the fractional logarithmic upper bound and its matching sharpness construction are new ingredients of the present argument before they are transferred to heat flux.

For the particularly transparent case $r=2$, Theorem 1.2 yields

$${\rho }_e\left(z\right)-{\rho }_i\left(z\right)\le C\left\{\begin{array}{cc}{\eta }_2\left(t\right),\hfill & 2\le N\le 4,\hfill \\ {\eta }_2\left(t\right){\left[1+{log}^{+}(C_{*}/{\eta }_2\left(t\right))\right]}^{1/2},\hfill & N=5,\hfill \\ {\eta }_2{\left(t\right)}^{{\displaystyle \frac{2(2+\alpha )}{N-1+2\alpha }}},\hfill & N\ge 6.\hfill \end{array}\right.$$

Thus the critical five-dimensional $L^2$ estimate carries a square-root logarithm. If $\alpha =1$, the subcritical exponent in dimensions $N\ge 6$ simplifies to $6/(N+1)$.

Finally, the quantitative transfer has a qualitative consequence stronger than exact discrete-time overdetermination. If $t_j\downarrow 0$ and

$${\delta }_r(t_j;\Omega )⟶0,$$

then (6) gives $H\equiv H_0$, and Alexandrov’s theorem implies that $\Omega$ is a ball. Thus exact spatial constancy of the heat flux at the observation times is unnecessary.

1.0.0.1. Organization.

Section 2 establishes the uniform short-time boundary-flux expansion. Section 3 converts the heat-flux defect into an $L^r$ mean-curvature defect. Section 4 proves the refined critical Alexandrov estimate and derives the complete heat-flux stability profile. Section 5 constructs the sharp subcritical, critical, and supercritical examples and transfers the corresponding profiles to the parabolic setting.

2. Uniform Short-Time Boundary Heat-Flux Asymptotics

Throughout, $\nu$ is the exterior unit normal and $n=-\nu$ the inward unit normal. The principal curvatures are taken with the convention for which $H=1/R$ on a sphere of radius R.

For a bounded connected domain $\Omega \subset {\mathbb{R}}^N$, let

$$u_{\Omega }\left(t\right)=e^{t{\Delta }_D}1$$

solve (2), and set

$$q_{\Omega }(y,t)={\partial }_{\nu }{u}_{\Omega }(y,t).$$

2.1. A Uniform Geometric Class

Fix $D,M,r_0>0$. We denote by ${\mathcal{D}}^{2,\alpha }(D,M,r_0)$ the class of bounded connected domains $\Omega \subset {\mathbb{R}}^N$ such that $diam\Omega \le D$ and, for every $Q\in \partial \Omega$, after a rigid motion sending Q to the origin, $\partial \Omega \cap B_{r_0}$ is exactly the graph

$$x_N={\phi }_Q\left(x^{\prime }\right)$$

of a function satisfying

$${\phi }_Q\left(0\right)=0,\nabla {\phi }_Q\left(0\right)=0,{\parallel {\phi }_Q\parallel }_{C^{2,\alpha }\left(B_{r_0}^{\prime }\right)}\le M,$$

with $\Omega \cap B_{r_0}$ lying on one fixed side of the graph.

All constants below may depend on $N,\alpha ,D,M,r_0$, but not on the individual domain. This uniform graph condition yields a common tubular radius $r_{*}>0$, uniform bounds for the principal curvatures, and a uniform $C^{0,\alpha }$ bound for H.

Let d be the signed distance, positive in $\Omega$. In the common tubular neighborhood,

$$x=\pi \left(x\right)+d\left(x\right)n\left(\pi \right(x\left)\right),\nabla d\left(x\right)=n\left(\pi \right(x\left)\right),$$

and, if $x=y+rn\left(y\right)$,

$$\Delta d\left(x\right)=-{\displaystyle \sum _{j=1}^{N-1}}{\displaystyle \frac{{\kappa }_j\left(y\right)}{1-r{\kappa }_j\left(y\right)}}.$$

(9)

Hence

$$\left|\Delta d\left(x\right)+(N-1)H\left(\pi \right(x\left)\right)\right|\le Cd\left(x\right).$$

(10)

2.2. Uniform Semigroup Smoothing

Lemma 2.1 

(Uniform Dirichlet gradient estimate). There exist $C,T_0>0$ such that for every $\Omega \in {\mathcal{D}}^{2,\alpha }(D,M,r_0)$, $f\in L^{\infty }\left(\Omega \right)$, and $0<s<T_0$,

$$\parallel \nabla e^{s{\Delta }_D}{f\parallel }_{L^{\infty }\left(\Omega \right)}\le C{s}^{-1/2}{\parallel f\parallel }_{L^{\infty }\left(\Omega \right)}.$$

(11)

Consequently,

$$\parallel {\partial }_{\nu }{e}^{s{\Delta }_D}{f\parallel }_{L^{\infty }(\partial \Omega )}\le C{s}^{-1/2}{\parallel f\parallel }_{L^{\infty }\left(\Omega \right)}.$$

(12)

Proof. 

Let $z(x,\sigma )=e^{\sigma {\Delta }_D}f\left(x\right)$. The maximum principle gives ${\parallel z\parallel }_{\infty }\le {\parallel f\parallel }_{\infty }$. At points a distance at least $c\sqrt{s}$ from the boundary, the standard interior parabolic gradient estimate on a cylinder of radius comparable with $\sqrt{s}$, followed by parabolic scaling, gives (11).

Near the boundary, choose a uniform graph chart, flatten the boundary, and rescale space by $\sqrt{s}$ and time by s. The transformed equation is uniformly parabolic on a unit half-cylinder, with homogeneous Dirichlet data and coefficient ellipticity and Lipschitz bounds controlled solely by the uniform $C^{1,1}$ character of the class. The local boundary gradient estimate therefore gives the same $C{s}^{-1/2}$ bound. A finite covering at the relevant local scale completes the proof. The positive-time boundary regularity gives the normal trace and (12). □

Remark 2.2. 

The estimate is consistent with standard boundary heat-kernel and gradient estimates on regular domains; see, for example, [11,12]. The local proof above is included to make the uniform dependence on the geometric class explicit.

2.3. Ambient Regularization and Boundary Layer

Choose $\vartheta \in C_c^{\infty }\left((-2r_{*},2r_{*})\right)$ with $\vartheta =1$ on $[-r_{*},r_{*}]$ and set

$$F\left(x\right)=\left\{\begin{array}{cc}\vartheta \left(d\right(x\left)\right)(N-1)H\left(\pi \right(x\left)\right),\hfill & x\in U_{2r_{*}},\hfill \\ 0,\hfill & x\notin U_{2r_{*}}.\hfill \end{array}\right.$$

Then, uniformly over the class,

$${\omega }_F\left(\rho \right)\le C{\rho }^{\alpha }.$$

(13)

For a standard mollifier, let $F_{\epsilon }={\rho }_{\epsilon }*F$ and put

$$h(x,t)=F_{\sqrt{t}}\left(x\right).$$

The cancellation identities for the differentiated convolution kernels give

$$\begin{array}{cc}\hfill {\parallel h-F\parallel }_{\infty }& \le C{t}^{\alpha /2},\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\parallel \nabla h\parallel }_{\infty }& \le C{t}^{(\alpha -1)/2},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \parallel D^2{h\parallel }_{\infty }+{\parallel {\partial }_{t}h\parallel }_{\infty }& \le C{t}^{-1+\alpha /2}.\hfill \end{array}$$

(16)

Set

$$V_0\left(\eta \right)=erfc(\eta /2),V_1\left(\eta \right)={\displaystyle \frac{\eta }{2}}erfc(\eta /2).$$

They satisfy

$$V_0^{\prime \prime }+{\displaystyle \frac{\eta }{2}}{V}_0^{\prime }=0,V_1^{\prime \prime }+{\displaystyle \frac{\eta }{2}}{V}_1^{\prime }-{\displaystyle \frac{1}{2}}{V}_1=V_0^{\prime },$$

with

$$V_0\left(0\right)=1,V_0^{\prime }\left(0\right)=-{\displaystyle \frac{1}{\sqrt{\pi }}},V_1\left(0\right)=0,V_1^{\prime }\left(0\right)={\displaystyle \frac{1}{2}},$$

and Gaussian decay of all derivatives at infinity.

Choose $\chi \in C_c^{\infty }\left([0,r_{*})\right)$ with $\chi =1$ near zero and define

$$V(x,t)=\chi \left(d\left(x\right)\right)\left[V_0\left({\displaystyle \frac{d\left(x\right)}{\sqrt{t}}}\right)+\sqrt{t}h(x,t)V_1\left({\displaystyle \frac{d\left(x\right)}{\sqrt{t}}}\right)\right].$$

(17)

Lemma 2.3 

(Uniform residual estimate). If $R=({\partial }_t-\Delta )V$, then for some uniform $C,T_1>0$,

$${\parallel R(·,t)\parallel }_{L^{\infty }\left(\Omega \right)}\le C{t}^{(\alpha -1)/2},0<t<T_1.$$

(18)

Proof. 

In the region where $\chi =1$, write $r=d\left(x\right)$, $\eta =r/\sqrt{t}$, and $a=\Delta d$. Direct differentiation gives

$$\begin{array}{cc}\hfill R=& -(a+h)t^{-1/2}{V}_0^{\prime }\left(\eta \right)-ah{V}_1^{\prime }\left(\eta \right)\hfill \\ \hfill & +\sqrt{t}({\partial }_{t}h-\Delta h)V_1\left(\eta \right)-2V_1^{\prime }\left(\eta \right)\nabla h·\nabla d.\hfill \end{array}$$

Since $F=(N-1)H\circ \pi$ in the inner collar, (10) and Gaussian decay give

$$|(a+F)t^{-1/2}{V}_0^{\prime }\left(\eta \right)|\le C.$$

Equation (14) gives

$$|(h-F)t^{-1/2}{V}_0^{\prime }\left(\eta \right)|\le C{t}^{(\alpha -1)/2}.$$

The term $ah{V}_1^{\prime }$ is uniformly bounded, while ()–() control the remaining terms by $C{t}^{(\alpha -1)/2}$. Derivatives of $\chi$ are supported a fixed positive distance from the boundary and are therefore exponentially small in $1/t$. This proves (18). □

Proposition 2.4 

(Uniform short-time boundary-flux expansion). There exist $C,t_{*}>0$, depending only on $N,\alpha ,D,M,r_0$, such that for every $\Omega \in {\mathcal{D}}^{2,\alpha }(D,M,r_0)$,

$$\underset{y\in \partial \Omega }{sup}\left|q_{\Omega }(y,t)+{\displaystyle \frac{1}{\sqrt{\pi t}}}-{\displaystyle \frac{N-1}{2}}H\left(y\right)\right|\le C{t}^{\alpha /2},0<t<t_{*}.$$

(19)

Proof. 

Set $v=1-u_{\Omega }$ and $w=v-V$. Then $w=0$ on the lateral boundary, $w(·,t)\to 0$ in $L^2$ as $t\downarrow 0$, and

$$w\left(t\right)=-{\int }_0^t{e}^{(t-s){\Delta }_D}R\left(s\right)\mathrm{d}s.$$

By Lemmas 2.1 and 2.3,

$$\begin{array}{cc}\hfill \parallel {\partial }_{\nu }{w(·,t)\parallel }_{\infty }& \le C{\int }_0^t{(t-s)}^{-1/2}{s}^{(\alpha -1)/2}\mathrm{d}s\hfill \\ \hfill & =C{t}^{\alpha /2}B\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1+\alpha }{2}}\right).\hfill \end{array}$$

At the boundary, $r=d$ increases in the inward direction and ${\partial }_{\nu }u={\partial }_{r}v$. Differentiating (17) at $r=0$ yields

$${\partial }_{r}V(y,t)=-{\displaystyle \frac{1}{\sqrt{\pi t}}}+{\displaystyle \frac{1}{2}}h(y,t).$$

Since $F\left(y\right)=(N-1)H\left(y\right)$ and (14) holds, the conclusion follows. □

3. From Heat-Flux Defects to Curvature Defects

For $1\le r\le \infty$ define

$$D_r\left(f\right)=\underset{c\in \mathbb{R}}{inf}{\parallel f-c\parallel }_{L^r(\partial \Omega )}.$$

The distance to the constants is invariant under addition of constants and is 1-Lipschitz:

$$|D_r\left(f\right)-D_r{\left(g\right)|\le \parallel f-g\parallel }_{L^r(\partial \Omega )}.$$

(20)

Lemma 3.1 

(Flux defect versus curvature distance). For every $1\le r\le \infty$ there exist uniform $C,t_{*}>0$ such that

$$\left|{\delta }_r(t;\Omega )-{\displaystyle \frac{N-1}{2}}{D}_r\left(H\right)\right|\le C{t}^{\alpha /2},0<t<t_{*}.$$

(21)

Proof. 

Proposition 2.4 gives

$$q_{\Omega }(·,t)=-{\displaystyle \frac{1}{\sqrt{\pi t}}}+{\displaystyle \frac{N-1}{2}}H+E_t,{\parallel E_t\parallel }_{\infty }\le C{t}^{\alpha /2}.$$

Apply (20) and use the uniform surface-area bound supplied by the fixed geometric class. □

Lemma 3.2 

(Minkowski normalization). Let $H_0=|\partial \Omega |/\left(N\right|\Omega \left|\right)$. For every $1\le r\le \infty$,

$$D_r\left(H\right)\le {\parallel H-H_0\parallel }_{L^r(\partial \Omega )}\le C{D}_r\left(H\right),$$

(22)

with C uniform on the geometric class.

Proof. 

The first inequality is immediate. For the converse, fix $z\in \Omega$. With the present inward-curvature/outward-normal convention, the first Minkowski identity is

$${\int }_{\partial \Omega }H\left(x\right)(x-z)·\nu \left(x\right)\mathrm{d}{S}_x=|\partial \Omega |,$$

(23)

while the divergence theorem gives

$${\int }_{\partial \Omega }(x-z)·\nu \left(x\right)\mathrm{d}{S}_x=N\left|\Omega \right|.$$

Hence for every $c\in \mathbb{R}$,

$$N\left|\Omega \right|(H_0-c)={\int }_{\partial \Omega }(H-c)(x-z)·\nu \mathrm{d}S.$$

By Hölder’s inequality and $\left|\right(x-z)·\nu |\le D$,

$$|H_0-{c\left|\right|\partial \Omega |}^{1/r}\le D{H}_0{\parallel H-c\parallel }_{L^r(\partial \Omega )}.$$

The fixed graph radius and norm, together with the diameter bound, give uniform upper and lower geometric bounds and in particular a uniform upper bound for $H_0$. Thus

$$\parallel H-H_0{\parallel }_{L^r}\le C{\parallel H-c\parallel }_{L^r}.$$

Taking the infimum over c proves the result. □

Proposition 3.3 

(Heat-flux defect controls curvature defect). For every $1\le r\le \infty$ there exist uniform $C,t_{*}>0$ such that

$$\parallel H-H_0{\parallel }_{L^r(\partial \Omega )}\le C\left[{\delta }_r(t;\Omega )+t^{\alpha /2}\right],0<t<t_{*}.$$

(24)

Proof. 

Combine Lemmas 3.1 and 3.2. □

Corollary 3.4 

(Asymptotic $L^r$ heat-flux rigidity). Suppose $t_j\downarrow 0$ and

$${\delta }_r(t_j;\Omega )⟶0$$

for some $1\le r\le \infty$. Then $H\equiv H_0$ on $\partial \Omega$, and Ω is a Euclidean ball.

Proof. 

Proposition 3.3 gives $\parallel H-H_0{\parallel }_{L^r}=0$. Since H is continuous, it is constant everywhere. Alexandrov’s theorem applies to each boundary component. With the present inward-curvature convention, an outer spherical component has positive H while a spherical boundary component surrounding a hole has negative H; hence no inner component can occur. Connectedness of $\Omega$ then yields a ball; see also the standard Alexandrov argument in [1]. □

4. Quantitative Rigidity from Short-Time Heat-Flux Defects

Set

$${\eta }_r(t;\Omega )={\delta }_r(t;\Omega )+t^{\alpha /2}.$$

(25)

For $r<(N-1)/2$, define

$${\tau }_{N,r,\alpha }={\displaystyle \frac{2+\alpha }{2+\alpha +{\displaystyle \frac{N-1-2r}{r}}}}.$$

(26)

$${\gamma }_N:={\displaystyle \frac{N-3}{N-1}}.$$

(27)

The critical case requires a refinement of the logarithmic step in the quantitative Alexandrov argument. We isolate the analytic input first.

Lemma 4.1 

(Critical Brézis–Gallouet–Wainger estimate). Let $d\ge 3$, $p=d/2$, $0<\beta <1$, and $M>0$. There exist constants $C,C_{*}>1$, depending only on $d,\beta$ and M, such that every $f\in W^{2,p}\left({\mathbb{S}}^d\right)\cap C^{0,\beta }\left({\mathbb{S}}^d\right)$ with

$${\parallel f\parallel }_{C^{0,\beta }\left({\mathbb{S}}^d\right)}\le M$$

and

$$A:={\parallel f\parallel }_{W^{2,p}\left({\mathbb{S}}^d\right)}\le 1$$

satisfies

$${\parallel f\parallel }_{L^{\infty }\left({\mathbb{S}}^d\right)}\le CA{\left[1+{log}^{+}(C_{*}/A)\right]}^{{\displaystyle \frac{p-1}{p}}}.$$

(28)

In particular, since $p=d/2$, the logarithmic exponent equals $(d-2)/d$.

Proof. 

Let $x_0\in {\mathbb{S}}^d$ be such that $|f\left(x_0\right){|=\parallel f\parallel }_{L^{\infty }\left({\mathbb{S}}^d\right)}$. After a rotation, work in a fixed stereographic chart sending $x_0$ to the origin. Let $\zeta$ be a fixed cutoff which is identically one near the origin and supported in the chart, and set

$$g=\zeta (f\circ {\iota }^{-1}),$$

extended by zero to ${\mathbb{R}}^d$. The fixed chart and cutoff give

$${\parallel g\parallel }_{{\dot{W}}^{2,p}\left({\mathbb{R}}^d\right)}\le CA,{\parallel g\parallel }_{{\dot{C}}^{\beta }\left({\mathbb{R}}^d\right)}+{\int }_{{\mathbb{R}}^d}{\displaystyle \frac{\left|g\right(y\left)\right|}{{(1+|y\left|\right)}^{d+1}}}dy\le C_M.$$

Choose $C_0\ge 1$ so that

$${\parallel g\parallel }_{{\dot{W}}^{2,p}\left({\mathbb{R}}^d\right)}\le C_{0}A,$$

and set $h=g/\left(C_{0}A\right)$. Then ${\parallel h\parallel }_{{\dot{W}}^{2,p}}\le 1$. Since the cutoff has fixed compact support and ${\parallel g\parallel }_{C^{0,\beta }}$ is bounded uniformly, the weighted $L^1$ and Hölder quantities entering Theorem 1.5 of Dao and Nguyen [3] satisfy

$$\underset{z\in {\mathbb{R}}^d}{sup}{\int }_{{\mathbb{R}}^d}{\displaystyle \frac{\left|h\right(y\left)\right|}{{(1+|z-y\left|\right)}^{\mu }}}dy+{\left[h\right]}_{{\dot{C}}^{\beta }}\le {\displaystyle \frac{C_M}{A}}$$

for some fixed admissible $\mu >d$. Because $2p=d$, their critical logarithmic inequality has power $(p-1)/p$, and hence

$$\left|h\left(0\right)\right|\le C{\left[1+{log}^{+}(C_M/A)\right]}^{(p-1)/p}.$$

Multiplying by $C_{0}A$ and adjusting the constants gives

$$|f\left(x_0\right)|=|g\left(0\right)|\le CA{\left[1+{log}^{+}(C_{*}/A)\right]}^{(p-1)/p}.$$

This proves (28). The power $(p-1)/p$ is the classical Brézis–Wainger critical exponent; see also [2,3]. □

Theorem 4.2 

(Refined critical Alexandrov stability). Let $N\ge 4$ and let $\Omega \in {\mathcal{D}}^{2,\alpha }(D,M,r_0)$. Set

$$p={\displaystyle \frac{N-1}{2}},{\epsilon }_H={\parallel H-H_0\parallel }_{L^p(\partial \Omega )}.$$

Then there exist class-dependent constants $C,C_{*}>1$ and a point $z\in \Omega$ such that

$${\rho }_e\left(z\right)-{\rho }_i\left(z\right)\le C{\epsilon }_H{\left[1+{log}^{+}(C_{*}/{\epsilon }_H)\right]}^{{\gamma }_N},$$

(29)

where ${\gamma }_N=(N-3)/(N-1)$.

Proof. 

We follow the nearly spherical reduction in the proof of Theorem 1.3 of Gonçalves da Silva and Poggesi [5], but replace their final critical interpolation step. If ${\epsilon }_H$ is below the class-dependent threshold in their Lemma 3.1, after a translation one may write

$$\partial \Omega =\left\{\left({\displaystyle \frac{1}{H_0}}+\omega \left(\theta \right)\right)\theta :\theta \in {\mathbb{S}}^{N-1}\right\},$$

with $\omega$ uniformly bounded in $C^{2,\alpha }$ and

$${\parallel \omega \parallel }_{W^{2,p}\left({\mathbb{S}}^{N-1}\right)}\le C{\parallel H-H_0\parallel }_{L^p(\partial \Omega )}=C{\epsilon }_H.$$

(30)

The estimate (30) is precisely the $W^{2,(N-1)/2}$ estimate used in their equation (3.26). Moreover,

$${\rho }_e\left(0\right)-{\rho }_i\left(0\right)\le 2{\parallel \omega \parallel }_{L^{\infty }\left({\mathbb{S}}^{N-1}\right)}.$$

Applying Lemma 4.1 with $d=N-1$ and $p=(N-1)/2$ gives

$${\parallel \omega \parallel }_{\infty }\le C{\epsilon }_H{\left[1+{log}^{+}(C_{*}/{\epsilon }_H)\right]}^{{\displaystyle \frac{N-3}{N-1}}},$$

after adjusting C and $C_{*}$. This proves (29) for small ${\epsilon }_H$.

If ${\epsilon }_H$ is above the fixed smallness threshold, the result follows from the diameter bound after enlarging C, since the right hand side is then bounded below by a positive class-dependent constant. □

Theorem 4.3 

(Quantitative rigidity from short-time heat flux). Let $\Omega \in {\mathcal{D}}^{2,\alpha }(D,M,r_0)$ and assume

$$r>1,r\ge {\displaystyle \frac{2N-2}{N+1}}.$$

There exist uniform constants $C,t_{*}>0$ and $C_{*}>1$ such that for every $0<t<t_{*}$ there is $z=z(\Omega ,t)\in \Omega$ satisfying

$${\rho }_e\left(z\right)-{\rho }_i\left(z\right)\le C{\Phi }_{N,r,\alpha }\left({\eta }_r(t;\Omega )\right),$$

(31)

where

$${\Phi }_{N,r,\alpha }\left(s\right)=\left\{\begin{array}{cc}s,\hfill & r>{\displaystyle \frac{N-1}{2}},\hfill \\ s{\left[1+{log}^{+}(C_{*}/s)\right]}^{{\gamma }_N},\hfill & r={\displaystyle \frac{N-1}{2}},\hfill \\ s^{{\tau }_{N,r,\alpha }},\hfill & r<{\displaystyle \frac{N-1}{2}},\hfill \end{array}\right.$$

(32)

Proof. 

By Proposition 3.3,

$$\parallel H-H_0{\parallel }_{L^r(\partial \Omega )}\le C_0{\eta }_r(t;\Omega ).$$

If $r>(N-1)/2$ or $r<(N-1)/2$, apply Theorem 1.5 of Gonçalves da Silva and Poggesi [5] with $k=2$. If $r=(N-1)/2$, apply Theorem 4.2. In each case the relevant profile is stable, up to a multiplicative constant, under the fixed dilation $s\mapsto C_{0}s$ after adjusting C and, in the critical case, $C_{*}$. This proves (31). □

Corollary 4.4 

($L^2$ heat-flux stability). Let

$${\eta }_2\left(t\right)=\underset{b\in \mathbb{R}}{inf}{\parallel {\partial }_{\nu }{u}_{\Omega }(·,t)-b\parallel }_{L^2(\partial \Omega )}+t^{\alpha /2}.$$

Then there exists $z\in \Omega$ such that

$${\rho }_e\left(z\right)-{\rho }_i\left(z\right)\le C\left\{\begin{array}{cc}{\eta }_2\left(t\right),\hfill & 2\le N\le 4,\hfill \\ {\eta }_2\left(t\right){\left[1+{log}^{+}(C_{*}/{\eta }_2\left(t\right))\right]}^{1/2},\hfill & N=5,\hfill \\ {\eta }_2{\left(t\right)}^{{\displaystyle \frac{2(2+\alpha )}{N-1+2\alpha }}},\hfill & N\ge 6.\hfill \end{array}\right.$$

If $\alpha =1$, the last exponent equals $6/(N+1)$ for $N\ge 6$. The critical $N=5$ logarithmic exponent is $1/2$.

Remark 4.5 

(Exact versus asymptotic overdetermination). If ${\partial }_{\nu }{u}_{\Omega }(·,t_j)$ is exactly constant for a sequence $t_j\downarrow 0$, then ${\delta }_r(t_j;\Omega )=0$. Corollary 3.4 therefore recovers exact discrete-time rigidity, but the hypothesis ${\delta }_r(t_j;\Omega )\to 0$ is strictly weaker.

5. Optimality of the Stability Profiles

We show that replacing the mean-curvature deviation by the short-time heat-flux defect does not improve any of the three stability regimes. In particular, we construct a logarithmic cap showing that the refined critical logarithmic exponent is optimal. The geometric perturbation parameter is denoted by $\epsilon$, while s denotes heat time.

5.1. Subcritical Regime

Assume

$$N\ge 4,{\displaystyle \frac{2N-2}{N+1}}\le r<{\displaystyle \frac{N-1}{2}},$$

(33)

and set

$$A_{N,r,\alpha }=2+\alpha +{\displaystyle \frac{N-1-2r}{r}}.$$

(34)

Then ${\tau }_{N,r,\alpha }=(2+\alpha )/A_{N,r,\alpha }$.

We use the optimality family of [5], specialized to $k=2$. Let

$${\phi }_0\left(x\right)=\sqrt{1-{\left|x\right|}^2},x\in B_1^{N-1},$$

and choose $\psi \in C_c^{\infty }\left(B_1^{N-1}\right)$ with $0\le \psi \le 1$ and $\psi \equiv 1$ on $B_{1/2}^{N-1}$. Define

$${\Psi }_{\epsilon }\left(x\right)=\psi (x/\epsilon ){\displaystyle \sum _{j=1}^{N-1}}{\left|x_j\right|}^{2+\alpha },{\phi }_{\epsilon }={\phi }_0+{\Psi }_{\epsilon },$$

and let ${\Gamma }_{\epsilon }$ be the sphere with its upper graph replaced by ${\phi }_{\epsilon }$ as in [5]; denote the enclosed domain by ${\Omega }_{\epsilon }$.

Lemma 5.1 

(Uniform regularity of the optimality family). For all sufficiently small ε, the domains ${\Omega }_{\epsilon }$ belong to one fixed class ${\mathcal{D}}^{2,\alpha }(D,M,r_0)$, and ${\Gamma }_{\epsilon }\to {\mathbb{S}}^{N-1}$ in $C^2$.

Proof. 

With

$$G\left(y\right)=\psi \left(y\right){\displaystyle \sum _{j=1}^{N-1}}{\left|y_j\right|}^{2+\alpha },$$

one has ${\Psi }_{\epsilon }\left(x\right)={\epsilon }^{2+\alpha }G(x/\epsilon )$. Hence

$$\parallel {\Psi }_{\epsilon }{\parallel }_{\infty }=O\left({\epsilon }^{2+\alpha }\right),\parallel \nabla {\Psi }_{\epsilon }{\parallel }_{\infty }=O\left({\epsilon }^{1+\alpha }\right),{\parallel D^2{\Psi }_{\epsilon }\parallel }_{\infty }=O\left({\epsilon }^{\alpha }\right).$$

Moreover,

$${\left[D^2{\Psi }_{\epsilon }\right]}_{C^{0,\alpha }}\le {\left[D^{2}G\right]}_{C^{0,\alpha }}$$

uniformly. Since the perturbation is supported in a shrinking neighborhood of one pole and the rest of the hypersurface is the unit sphere, a fixed finite atlas yields the required uniform class. □

Let

$${\mathcal{A}}_{\epsilon }=\underset{z\in {\Omega }_{\epsilon }}{inf}\left[{\rho }_e\left(z\right)-{\rho }_i\left(z\right)\right].$$

The construction in [5] gives

$${\mathcal{A}}_{\epsilon }\ge c{\epsilon }^{2+\alpha },$$

(35)

and, by their estimates together with Remark 4.1,

$$\parallel H_{\epsilon }-H_{0,\epsilon }{\parallel }_{L^r\left({\Gamma }_{\epsilon }\right)}\asymp {\epsilon }^{A_{N,r,\alpha }}.$$

(36)

Lemma 3.2 therefore gives

$$D_r\left(H_{\epsilon }\right)\asymp {\epsilon }^{A_{N,r,\alpha }}.$$

(37)

Choose

$$p>{\displaystyle \frac{2A_{N,r,\alpha }}{\alpha }},s_{\epsilon }={\epsilon }^p.$$

(38)

Then $s_{\epsilon }^{\alpha /2}=o\left({\epsilon }^{A_{N,r,\alpha }}\right)$. By Lemma 3.1, uniformly in $\epsilon$,

$${\delta }_r(s_{\epsilon };{\Omega }_{\epsilon })\asymp {\epsilon }^{A_{N,r,\alpha }},$$

(39)

and the same is true for ${\eta }_r(s_{\epsilon };{\Omega }_{\epsilon })$.

Theorem 5.2 

(Sharpness in the subcritical regime). Under (33), the exponent ${\tau }_{N,r,\alpha }$ in Theorem 4.3 cannot be replaced uniformly by any larger power exponent.

Proof. 

Equations (35) and (39), together with

$$A_{N,r,\alpha }{\tau }_{N,r,\alpha }=2+\alpha ,$$

give

$${\mathcal{A}}_{\epsilon }\ge c{\delta }_r{(s_{\epsilon };{\Omega }_{\epsilon })}^{{\tau }_{N,r,\alpha }}.$$

If an estimate with an exponent $\beta >{\tau }_{N,r,\alpha }$ held uniformly, then

$$c{\epsilon }^{2+\alpha }\le C{\epsilon }^{A_{N,r,\alpha }\beta },$$

which is impossible because $A_{N,r,\alpha }\beta >2+\alpha$. □

5.2. Critical Regime

Assume $N\ge 4$ and set

$$d=N-1,p={\displaystyle \frac{d}{2}}={\displaystyle \frac{N-1}{2}},{\gamma }_N=1-{\displaystyle \frac{1}{p}}={\displaystyle \frac{N-3}{N-1}}.$$

(40)

We now construct a uniformly $C^{2,\alpha }$ family for which the logarithmic power ${\gamma }_N$ in Theorem 4.2, and hence in Theorem 4.3, is attained.

Choose a geodesic normal coordinate chart $\kappa :B_{3\varrho }^d\left(0\right)\to {\mathbb{S}}^d$ centered at the north pole, where $\varrho >0$ is fixed and sufficiently small. Let $\chi \in C_c^{\infty }\left(B_{2\varrho }^d\right)$ satisfy $0\le \chi \le 1$ and $\chi \equiv 1$ on $B_{\varrho }^d$. For $0<\epsilon \ll 1$, define

$$g_{\epsilon }\left(x\right)=\chi \left(x\right)log{\displaystyle \frac{3\varrho }{\sqrt{{\left|x\right|}^2+{\epsilon }^2}}},a_{\epsilon }={\epsilon }^{2+\alpha },$$

(41)

and let ${\omega }_{\epsilon }:{\mathbb{S}}^d\to [0,\infty )$ be given by

$${\omega }_{\epsilon }\left(\kappa \left(x\right)\right)=a_{\epsilon }{g}_{\epsilon }\left(x\right)\mathrm{on}\kappa \left(B_{3\varrho }\right),{\omega }_{\epsilon }=0\mathrm{outside}\mathrm{the}\mathrm{chart}.$$

The cutoff and the factor $3\varrho$ are chosen so that $g_{\epsilon }\ge 0$ on its support for all small $\epsilon$. Define the radial graph

$${\Gamma }_{\epsilon }=\{(1+{\omega }_{\epsilon }\left(\theta \right))\theta :\theta \in {\mathbb{S}}^d\},$$

(42)

and let ${\Omega }_{\epsilon }$ be the enclosed domain.

Lemma 5.3 

(Uniform regularity of the logarithmic caps). The family $\left\{{\Omega }_{\epsilon }\right\}$ belongs, for all sufficiently small ε, to one fixed class ${\mathcal{D}}^{2,\alpha }(D,M,r_0)$, and ${\Gamma }_{\epsilon }\to {\mathbb{S}}^d$ in $C^2$.

Proof. 

On the support of $\chi$, derivatives of the uncut logarithm satisfy

$$|D^m{g}_{\epsilon }\left(x\right)|\le C_m{\left(\right|x|+\epsilon )}^{-m},m=1,2,3,$$

up to uniformly bounded cutoff terms. Hence

$$\parallel {\omega }_{\epsilon }{\parallel }_{\infty }\le C{a}_{\epsilon }log(1/\epsilon ),\parallel \nabla {\omega }_{\epsilon }{\parallel }_{\infty }\le C{a}_{\epsilon }{\epsilon }^{-1},{\parallel D^2{\omega }_{\epsilon }\parallel }_{\infty }\le C{a}_{\epsilon }{\epsilon }^{-2}=C{\epsilon }^{\alpha }.$$

Thus ${\omega }_{\epsilon }\to 0$ in $C^2$. To control the Hölder seminorm, use

$$|D^2{\omega }_{\epsilon }\left(x\right)-D^2{\omega }_{\epsilon }\left(y\right)|\le Cmin\{{\epsilon }^{\alpha },{\epsilon }^{\alpha -1}|x-y|\}.$$

If $|x-y|\le \epsilon$, divide the second bound by ${|x-y|}^{\alpha }$; if $|x-y|\ge \epsilon$, divide the first one. In both cases the resulting quantity is bounded uniformly. Fixed-coordinate changes on the sphere preserve these estimates up to constants. Therefore ${\left[D^2{\omega }_{\epsilon }\right]}_{C^{0,\alpha }}\le C$ uniformly, which proves the claim. □

Let

$$L_{\epsilon }=log(1/\epsilon ),M_{\epsilon }=\underset{{\mathbb{S}}^d}{max}{\omega }_{\epsilon }.$$

From (41),

$$M_{\epsilon }\asymp a_{\epsilon }{L}_{\epsilon }.$$

(43)

Lemma 5.4 

(Annular deviation of the logarithmic caps). For all sufficiently small ε,

$$\underset{z\in {\Omega }_{\epsilon }}{inf}[{\rho }_e\left(z\right)-{\rho }_i\left(z\right)]\asymp a_{\epsilon }{L}_{\epsilon }.$$

(44)

Proof. 

Taking $z=0$ and using that the perturbation is nonnegative gives the upper bound

$${\rho }_e\left(0\right)-{\rho }_i\left(0\right)=M_{\epsilon }.$$

For the converse, let $w_{\epsilon }\left(e\right)$ denote the directional width of ${\Omega }_{\epsilon }$ in direction e. Choose $\varrho$ so small that the support of the cap lies in a fixed neighborhood of the north pole whose projection onto a fixed tangential direction $e_1$ has absolute value at most $1/2$. Since $M_{\epsilon }\to 0$, for all sufficiently small $\epsilon$ one has $(1+M_{\epsilon })/2<1$. Hence every perturbed point in the cap satisfies

$$|(1+{\omega }_{\epsilon }\left(\theta \right))\theta ·e_1|<1,$$

whereas the unperturbed points $\pm e_1$ lie outside the support of the cap. Thus the supporting points in the $e_1$ direction remain $\pm e_1$ and, exactly,

$$w_{\epsilon }\left(e_1\right)=2.$$

In the north–south direction $e_{d+1}$, the north pole is displaced outward by $M_{\epsilon }$ while the south pole is unchanged, so

$$w_{\epsilon }\left(e_{d+1}\right)=2+M_{\epsilon }.$$

If $B_{{\rho }_i\left(z\right)}\left(z\right)\subset {\Omega }_{\epsilon }\subset B_{{\rho }_e\left(z\right)}\left(z\right)$, then $2{\rho }_i\left(z\right)\le w_{\epsilon }\left(e\right)\le 2{\rho }_e\left(z\right)$ for every e. Hence

$$2[{\rho }_e\left(z\right)-{\rho }_i\left(z\right)]\ge w_{\epsilon }\left(e_{d+1}\right)-w_{\epsilon }\left(e_1\right)=M_{\epsilon }.$$

Taking the infimum over z and using (43) proves the result. □

We next estimate the mean-curvature defect. Since the critical sharpness argument depends on the precise linearization, we record the radial-graph formula and the remainder estimate explicitly.

Lemma 5.5 

(Mean curvature of a radial graph). Let $d\ge 2$ and let

$$X\left(\theta \right)=\rho \left(\theta \right)\theta ,\rho =1+\omega >0,\theta \in {\mathbb{S}}^d.$$

All covariant derivatives in this lemma are taken with respect to the standard metric σ on ${\mathbb{S}}^d$. Put

$$q={|\nabla \rho |}^2,W={({\rho }^2+q)}^{1/2}.$$

With the outward unit normal and with normalized mean curvature equal to 1 on the unit sphere, one has the exact formula

$$H_{\rho }={\displaystyle \frac{1}{dW}}\left[d-{\displaystyle \frac{\Delta \rho }{\rho }}+{\displaystyle \frac{{|\nabla \rho |}^2}{W^2}}+{\displaystyle \frac{{\nabla }^2\rho (\nabla \rho ,\nabla \rho )}{\rho W^2}}\right].$$

(45)

Consequently, there exist $c_0,C>0$, depending only on d, such that whenever ${\parallel \omega \parallel }_{C^1\left({\mathbb{S}}^d\right)}\le c_0$,

$$H_{\omega }-1=-{\displaystyle \frac{1}{d}}({\Delta }_{{\mathbb{S}}^d}+d)\omega +\mathcal{R}\left(\omega \right),$$

(46)

where, for every $1\le p\le \infty$,

$${\parallel \mathcal{R}\left(\omega \right)\parallel }_{L^p\left({\mathbb{S}}^d\right)}\le {C\parallel \omega \parallel }_{C^1\left({\mathbb{S}}^d\right)}{\parallel \omega \parallel }_{W^{2,p}\left({\mathbb{S}}^d\right)}.$$

(47)

In particular, the same estimate holds with ${\parallel \omega \parallel }_{C^2}$ on the right-hand side.

Proof. 

Choose local coordinates on ${\mathbb{S}}^d$ and write subscripts for covariant derivatives. The tangent vectors of the radial graph are

$$X_i={\rho }_i\theta +\rho e_i,$$

so its induced metric and inverse metric are

$$g_{ij}={\rho }^2{\sigma }_{ij}+{\rho }_i{\rho }_j,g^{ij}={\rho }^{-2}\left({\sigma }^{ij}-{\displaystyle \frac{{\rho }^i{\rho }^j}{W^2}}\right).$$

(48)

The outward unit normal is

$$\nu ={\displaystyle \frac{\rho \theta -\nabla \rho }{W}}.$$

(49)

Using

$$D_i(\nabla \rho )={\nabla }_i\nabla \rho -{\rho }_i\theta$$

in the ambient Euclidean connection gives, with the convention $h_{ij}=\langle D_i\nu ,X_j\rangle$,

$$h_{ij}={\displaystyle \frac{1}{W}}\left({\rho }^2{\sigma }_{ij}+2{\rho }_i{\rho }_j-\rho {\rho }_{ij}\right).$$

(50)

For constant $\rho =R$, this gives $h_{ij}=R{\sigma }_{ij}$ and hence positive curvature $1/R$, confirming the sign convention.

Since $H_{\rho }=d^{-1}{g}^{ij}{h}_{ij}$, contracting (48) and (50) yields

$$\begin{array}{cc}\hfill d{H}_{\rho }& ={\displaystyle \frac{1}{{\rho }^{2}W}}\left[d{\rho }^2+2q-\rho \Delta \rho -{\displaystyle \frac{{\rho }^{2}q+2q^2-\rho {\nabla }^2\rho (\nabla \rho ,\nabla \rho )}{W^2}}\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{W}}\left[d-{\displaystyle \frac{\Delta \rho }{\rho }}+{\displaystyle \frac{q}{W^2}}+{\displaystyle \frac{{\nabla }^2\rho (\nabla \rho ,\nabla \rho )}{\rho W^2}}\right],\hfill \end{array}$$

which is (45).

Now set $\rho =1+\omega$. If ${\parallel \omega \parallel }_{C^1}\le c_0$ with $c_0$ small, then $\rho$ and W are bounded above and below by positive constants. Expanding only the smooth coefficients in (45), while keeping the second derivatives of $\omega$ unexpanded, gives the pointwise estimate

$$\left|H_{\omega }-1+{\displaystyle \frac{1}{d}}(\Delta \omega +d\omega )\right|\le C{\parallel \omega \parallel }_{C^1}\left(\left|\omega \right|+|\nabla \omega |+|{\nabla }^2\omega |\right).$$

(51)

Indeed, the coefficient of $\Delta \omega$ differs from $-1/d$ by $O\left(\right|\omega |+|\nabla \omega \left|\right)$, the zeroth-order factor $W^{-1}$ differs from $1-\omega$ by a quadratic quantity, and the last two terms in (45) are at least quadratic in $(\omega ,\nabla \omega )$, with the Hessian term carrying two gradient factors. Taking the $L^p$ norm of (51) proves (47). □

Lemma 5.6 

(Critical curvature scale). Let $H_{\epsilon }$ be the normalized mean curvature of ${\Gamma }_{\epsilon }$ and $H_{0,\epsilon }=|{\Gamma }_{\epsilon }|/(N|{\Omega }_{\epsilon }\left|\right)$. Then

$$\parallel H_{\epsilon }-H_{0,\epsilon }{\parallel }_{L^p\left({\Gamma }_{\epsilon }\right)}\asymp a_{\epsilon }{L}_{\epsilon }^{1/p}=a_{\epsilon }{L}_{\epsilon }^{2/d}.$$

(52)

Proof. 

In the region where $\chi \equiv 1$, writing $r=\left|x\right|$ gives

$${\displaystyle \frac{d}{dr}}\left(-{\displaystyle \frac{1}{2}}log(r^2+{\epsilon }^2)\right)=-{\displaystyle \frac{r}{r^2+{\epsilon }^2}}$$

and

$${\Delta }_{{\mathbb{R}}^d}\left(-{\displaystyle \frac{1}{2}}log(r^2+{\epsilon }^2)\right)=-{\displaystyle \frac{(d-2)r^2+d{\epsilon }^2}{{(r^2+{\epsilon }^2)}^2}}.$$

(53)

Thus, on the annulus $C\epsilon <r<{\varrho }_1$ with fixed sufficiently small ${\varrho }_1$, the Euclidean second derivatives and Laplacian are of size $r^{-2}$. In geodesic normal coordinates on the sphere, metric errors are lower order; after decreasing ${\varrho }_1$, they do not affect the two-sided estimate for $({\Delta }_{{\mathbb{S}}^d}+d)g_{\epsilon }$ on this annulus. Since $2p=d$,

$${\int }_{C\epsilon }^{{\varrho }_1}{r}^{-2p}{r}^{d-1}dr\asymp L_{\epsilon }.$$

The inner ball $r\lesssim \epsilon$ and the cutoff region contribute only $O\left(1\right)$ to the corresponding pth power. Consequently,

$$\parallel {\omega }_{\epsilon }{\parallel }_{W^{2,p}\left({\mathbb{S}}^d\right)}\asymp a_{\epsilon }{L}_{\epsilon }^{1/p}$$

(54)

and

$$\parallel ({\Delta }_{{\mathbb{S}}^d}+d){\omega }_{\epsilon }{\parallel }_{L^p}\asymp a_{\epsilon }{L}_{\epsilon }^{1/p}.$$

By Lemma 5.3, $\parallel {\omega }_{\epsilon }{\parallel }_{C^2}=O\left({\epsilon }^{\alpha }\right)=o\left(1\right)$, so (47) is negligible relative to (54). The radial parametrization has a Jacobian uniformly comparable with one, so $L^p$ norms on ${\Gamma }_{\epsilon }$ and on ${\mathbb{S}}^d$ are equivalent uniformly. Hence

$$\parallel H_{\epsilon }{-1\parallel }_{L^p\left({\Gamma }_{\epsilon }\right)}\asymp a_{\epsilon }{L}_{\epsilon }^{1/p}.$$

(55)

It remains to replace 1 by $H_{0,\epsilon }$. The radial volume and area formulas give

$$N|{\Omega }_{\epsilon }|={\int }_{{\mathbb{S}}^d}{(1+{\omega }_{\epsilon })}^{N}d\sigma ,$$

$$|{\Gamma }_{\epsilon }|={\int }_{{\mathbb{S}}^d}{(1+{\omega }_{\epsilon })}^d\sqrt{1+{\displaystyle \frac{|\nabla {\omega }_{\epsilon }{|}^2}{{(1+{\omega }_{\epsilon })}^2}}}d\sigma .$$

The logarithmic singularity is integrable in dimension $d\ge 3$, so $\int {\omega }_{\epsilon }d\sigma =O\left(a_{\epsilon }\right)$, while $\int ({\omega }_{\epsilon }^2+|\nabla {\omega }_{\epsilon }{|}^2)d\sigma =O\left(a_{\epsilon }^2\right)$. Therefore

$$H_{0,\epsilon }-1=O\left(a_{\epsilon }\right)=o\left(a_{\epsilon }{L}_{\epsilon }^{1/p}\right).$$

Combining this with (55) proves (52). □

Theorem 5.7 

(Sharpness in the critical regime). Let $N\ge 4$, $p=(N-1)/2$, and ${\gamma }_N=(N-3)/(N-1)$. There exist uniformly $C^{2,\alpha }$ domains ${\Omega }_{\epsilon }\to B_1$ in $C^2$ and times $s_{\epsilon }\downarrow 0$ such that

$$\underset{z\in {\Omega }_{\epsilon }}{inf}[{\rho }_e\left(z\right)-{\rho }_i\left(z\right)]\asymp {\delta }_p(s_{\epsilon };{\Omega }_{\epsilon }){\left[log(1/{\delta }_p(s_{\epsilon };{\Omega }_{\epsilon }))\right]}^{{\gamma }_N}.$$

(56)

Consequently, the logarithmic exponent ${\gamma }_N$ in the critical branch of Theorem 4.3 cannot be replaced by any smaller exponent.

Proof. 

By Lemma 3.2 and Lemma 5.6, uniformly along the family,

$$D_p\left(H_{\epsilon }\right)\asymp a_{\epsilon }{L}_{\epsilon }^{1/p}.$$

(57)

Choose

$$s_{\epsilon }={\epsilon }^m,{\displaystyle \frac{m\alpha }{2}}>2+\alpha .$$

Then

$$s_{\epsilon }^{\alpha /2}=o\left(a_{\epsilon }{L}_{\epsilon }^{1/p}\right).$$

Lemma 3.1 and (57) therefore yield

$${\delta }_p(s_{\epsilon };{\Omega }_{\epsilon })\asymp a_{\epsilon }{L}_{\epsilon }^{1/p}.$$

(58)

On the other hand, Lemma 5.4 gives

$$\underset{z}{inf}[{\rho }_e\left(z\right)-{\rho }_i\left(z\right)]\asymp a_{\epsilon }{L}_{\epsilon }.$$

Since $a_{\epsilon }={\epsilon }^{2+\alpha }$,

$$log{\displaystyle \frac{1}{{\delta }_p(s_{\epsilon };{\Omega }_{\epsilon })}}=(2+\alpha )L_{\epsilon }+O(log{L}_{\epsilon }),$$

and hence this logarithm is comparable with $L_{\epsilon }$. Finally,

$$1-{\displaystyle \frac{1}{p}}={\displaystyle \frac{N-3}{N-1}}={\gamma }_N,$$

which proves (56).

If the critical estimate held uniformly with a logarithmic power $\beta <{\gamma }_N$, the right hand side along this family would be $o\left(a_{\epsilon }{L}_{\epsilon }\right)$, contradicting Lemma 5.4. Thus ${\gamma }_N$ is optimal. □

5.3. Supercritical Regime

Assume $r>(N-1)/2$. Consider the ellipsoids

$$E_{\epsilon }=\left\{x\in {\mathbb{R}}^N:{\displaystyle \frac{x_1^2}{{(1+\epsilon )}^2}}+{\displaystyle \sum _{j=2}^N}{x}_j^2<1\right\}.$$

They form a fixed smooth geometric class. As noted in [5], ellipsoids already test the sharpness of the linear Alexandrov profile; expanding this family at the unit ball gives

$$\underset{z}{inf}[{\rho }_e\left(z\right)-{\rho }_i\left(z\right)]\asymp \epsilon ,D_r\left(H_{E_{\epsilon }}\right)\asymp \epsilon .$$

Choose $s_{\epsilon }={\epsilon }^p$ with $p>2/\alpha$. Then Lemma 3.1 gives

$${\delta }_r(s_{\epsilon };E_{\epsilon })\asymp \epsilon .$$

Thus the linear exponent 1 in Theorem 4.3 is optimal.

Remark 5.8 

(Interpretation of the sharpness transfer). Along the sharp families above, the heat time can be chosen so that

$$t^{\alpha /2}=o\left(D_r\left(H\right)\right),$$

and hence

$${\delta }_r(t;\Omega )={\displaystyle \frac{N-1}{2}}{D}_r\left(H\right)+o\left(D_r\left(H\right)\right).$$

Thus the leading geometric defect is retained, to first order in the relevant norm, by the short-time boundary heat-flux measurement. In the critical family the same statement holds at the scale $D_p\left(H\right)\asymp a_{\epsilon }{L}_{\epsilon }^{1/p}$, which preserves the sharp logarithmic correction as well.