Generic uniqueness of expanders with vanishing relative entropy

Abstract

We define a relative entropy for two self-similarly expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same cone at infinity. Adapting work of White (Indiana Univ Math J 36(3):567–602, 1987) and using recent results of Bernstein (Asymptotic structure of almost eigenfunctions of drift laplacians on conical ends) and Bernstein-Wang (The space of asymptotically conical self-expanders of mean curvature flow), we show that expanders with vanishing relative entropy are unique in a generic sense. This also implies that generically locally entropy minimising expanders are unique.

Appendices

Appendix A. Geometry of normal graphs

Let again $\Sigma _0,\Sigma _1$ be two expanders asymptotic to the cone $C(\Gamma )$, and assume ${\bar{E}}_{1,R_0} = \text {graph}_{{\bar{E}}_{0,R_0}}(u)$ with $u : {\bar{E}}_{0,R_0} \rightarrow {\mathbb {R}}$. Let $p \in \Sigma _0$ and choose a local parametrisation F, parametrising an open neighbourhood U of p in $\Sigma _0$ such that $F(0)=p$. We can assume that $g_{ij}=\langle \partial _i F, \partial _j F \rangle$ satisfies

$$\begin{aligned} g_{ij}\big |_{x=0} = \delta _{ij} \text { and } \partial _kg_{ij}|_{x=0} = 0\, . \end{aligned}$$

For simplicity we can furthermore assume that the second fundamental form $(h_{ij})$ is diagonalised at p with eigenvalues $\lambda _1,\ldots , \lambda _n$. A direct calculation, see [10, (2.27)], yields that the normal vector $\nu _1(q)$, where $q = p + u(p) \nu _0(q)$, is co-linear to the vector

$$\begin{aligned} N = - \sum _{i=1}^n \frac{\partial _i u}{1 - \lambda _i u} \partial _iF\bigg |_{x=0} + \nu _0(p)\ . \end{aligned}$$

Denoting the shape operator $S = (h^i_{\ j})$ we see that thus in coordinate free notation

$$\begin{aligned} \nu _1(q) = v^{-1} \left( - (\text {Id} - uS)^{-1}\nabla _{\Sigma _0} u + \nu _0\right) (p)\, , \end{aligned}$$

(62)

where $v:=(1 + |(\text {Id} - uS)^{-1}(\nabla _{\Sigma _0} u)|^2)^{\frac{1}{2}}$. This implies

$$\begin{aligned} \langle q, \nu _1(q) \rangle = v^{-1} \left( u + \langle p, \nu _0(p) \rangle - \big \langle p, (\text {Id} - uS)^{-1}\nabla _{\Sigma _0} u \big \rangle \right) \, . \end{aligned}$$

(63)

For the induced metric ${{\tilde{g}}}$ one obtains in the above coordinates at p, again see [10, (2.32)],

$$\begin{aligned} {{\tilde{g}}}_{ij} = (1-\lambda _iu)(1-\lambda _ju) \delta _{ij} + \partial _i u \partial _j u \end{aligned}$$

(64)

which implies

$$\begin{aligned} {{\tilde{g}}}^{ij} = \frac{\delta ^{ij}}{(1-\lambda _i u)(1-\lambda _j u)} - v^{-2}\frac{\partial _i u}{(1-\lambda _i u)^2}\frac{\partial _j u}{(1-\lambda _j u)^2}\, . \end{aligned}$$

(65)

Furthermore, from [10, (2.30)] we have

$$\begin{aligned} {{\tilde{h}}}_{ij}= & {} \langle \partial ^2_{ij} {{\tilde{F}}}, \nu _N \rangle \nonumber \\= & {} v^{-1} \Big ( \frac{\lambda _i}{1-\lambda _i u} \partial _i u \partial _ju + \frac{\lambda _j}{1-\lambda _j u} \partial _i u \partial _ju\nonumber \\&+ \sum _k \frac{u}{1-\lambda _k u}\, \partial _k u \,\partial _i h_{jk} + h_{ij} - \lambda _i\lambda _j u\, \delta _{ij} + \partial ^2_{ij} u\Big )\, . \end{aligned}$$

(66)

Since $H_{\Sigma _1}(p) = {{\tilde{g}}}^{ij}(p) {{\tilde{h}}}_{ij}(p)$ we see that

$$\begin{aligned} H_{\Sigma _1}(p)= & {} \Delta _{\Sigma _0} u + u |A_{\Sigma _0}|^2 + H_{\Sigma _0}(q) + Q(x,u,\nabla ^{\Sigma _0} u , \nabla _{\Sigma _0}^2 u)\, , \end{aligned}$$

(67)

where $Q(x,u,\nabla ^{\Sigma _0} u , \nabla _{\Sigma _0}^2 u)$ is quadratic in $u, \nabla ^{\Sigma _0} u, \nabla _{\Sigma _0}^2u$.

Appendix B. Interpolation inequalities

We recall the following standard interpolation inequalities in multiplicative form.

Lemma B.1

Suppose that $u \in C^{k}(B_{2})$, then for $j< k$,

$$\begin{aligned} \Vert D^{j} u \Vert _{C^{0}(B_{1})} \le C \Vert u\Vert _{C^{0}(B_{2})}^{1-\frac{j}{k}} \Vert D^{k} u \Vert _{C^{0}(B_{2})}^{\frac{j}{k}} \end{aligned}$$

for $C=C(n,k)$. Similarly, if $u \in C^{k,\alpha }(B_{2})$, then for $j+\beta < k+\alpha$,

$$\begin{aligned} {[ D^{j} u ]}_{\beta ;B_{1}} \le C \Vert u\Vert _{C^{0}(B_{2})}^{1-\frac{j+\beta }{k+\alpha }} [ D^{k} u ]_{\alpha ; B_{2}}^{\frac{j+\beta }{k+\alpha }} \end{aligned}$$

for $C=C(n,k,\alpha ,\beta )$.

These follow in a similar manner to the linear inequalities given in [5, Lemma 6.32], except in the proof one should optimize with respect to the parameter $\mu$ rather than just choosing $\mu$ sufficiently small. Alternatively, see [6, Lemma A.2].