The Second Inner Variation of Energy and the Morse Index of Limit Interfaces

Abstract

In this article, we study the second variation of the energy functional associated to the Allen–Cahn equation on closed manifolds. Extending well-known analogies between the gradient theory of phase transitions and the theory of minimal hypersurfaces, we prove the upper semicontinuity of the eigenvalues of the stability operator and consequently obtain upper bounds for the Morse index of limit interfaces which arise from solutions with bounded energy and index without assuming any multiplicity or orientability condition on these hypersurfaces. This extends some recent results of Le (Indiana Univ Math J 60:1843–1856, 2011; J Math Pures Appl 103:1317–1345, 2015)) and Hiesmayr (arXiv:1704.07738 preprint [math.DG], 2017).

Appendix

Let M be a complete Riemannian manifold and let $\varSigma \subset M$ be a hypersurface. Given $\delta >0$ denote

$$\begin{aligned} B_\delta (N\varSigma ) = \{ v \in TM|_{\varSigma } : v \perp T\varSigma , |v|<\delta \}, \end{aligned}$$

and let $\pi :B_\delta (N\varSigma ) \rightarrow \varSigma$ be the projection map. Assume we can find $\delta >0$ such that the normal exponential map$F:B_\delta (N\varSigma ) \rightarrow M$ given by $F(v_x)=\exp _xv$ is a diffeomorphism onto an open set $U_\delta \subset M.$ Under these conditions, we have

Lemma

Every compactly supported smooth normal vector field X defined on $\varSigma$ admits a smooth extension ${\tilde{X}}$ to M such that $\nabla _v {\tilde{X}}=0$ for all $v \in N\varSigma \subset TM.$ Moreover, we can assume that ${\tilde{X}}$ vanishes outside of a neighborhood $V \subset U_\delta$ of $\varSigma .$

Proof

Intuitively, we will define ${\tilde{X}}$ at some $x \in U_\delta$ via parallel transport along $\gamma :[0,r] \rightarrow M,$ the unique geodesic which is normal to $\varSigma$ at $\gamma (0) \in \varSigma$ and satisfies $|\gamma '|=1$ and $\gamma (r)=x.$ Then, we may use a cutoff function to extend ${\tilde{X}}$ to M.

To make this idea precise, write $p=\pi \circ F^{-1}:U_\delta \rightarrow \varSigma ,$ and for every $x \in U_\delta$ let

$$\begin{aligned} {\hat{X}}|_x = \dfrac{\mathrm{d}}{\mathrm{d}t}\bigg |_{t=0} \exp _{p(x)}(F^{-1}(x) + tX|_{p(x)}) = \mathrm{d}\exp _{p(x)}(F^{-1}(x))\,(X|_{p(x)}). \end{aligned}$$

Clearly, ${\hat{X}}$ is a vector field on $U_\delta$ which is as regular as X,  and if $x \in \varSigma$ then $F^{-1}(x)=0 \in T_xM$ and $p(x)=x,$ so ${\hat{X}}|_x = X|_x.$ Moreover, since the fibers of $N\varSigma$ have dimension 1,  whenever $x \in U_\delta$ is such that $X|_{p(x)} \ne 0$, we have

$$\begin{aligned} F^{-1}(x)=s(x)X|_{p(x)} \quad \text{ for } \text{ some } \quad s(x) \in \mathbb {R}, \end{aligned}$$

with $s(x)=0$ for $x \in \varSigma .$ Hence ${\hat{X}}|_{x}$ is the tangent vector to the normal geodesic

$$\begin{aligned} \gamma (t)=\exp _{p(x)}(tX|_{p(x)}) \end{aligned}$$

at $t=s(x)$ and for all $x \in \varSigma$ in which X does not vanish we have

$$\begin{aligned} 0 = \dfrac{\mathrm{D}\gamma '}{\mathrm{d}t}\bigg |_{t=0} = \nabla _{X|_x}\hat{X}. \end{aligned}$$

This proves that

$$\begin{aligned} \nabla _v {\hat{X}}|_x=0 \quad \text{ for } \text{ all } \quad x \in \varSigma \quad \text{ s.t. } \quad X|_x \ne 0. \end{aligned}$$

Finally if $x \in \varSigma$ and $X|_x=0$ then ${\hat{X}}$ vanishes along the points in $p^{-1}(x) \cap U_\delta ,$ namely, the geodesic $\gamma (t)=\exp _x(t \mathbf {n})$ for an unit $\mathbf {n} \in N_x\varSigma$ and $|t|<\delta .$ We conclude thus

$$\begin{aligned} 0 = \dfrac{\mathrm{D} ({\hat{X}}|_\gamma )}{\mathrm{d}t}\bigg |_{t=0} = \nabla _{\gamma '(0)}{\hat{X}} = \nabla _{\mathbf {n}} {\hat{X}} \end{aligned}$$

and $\nabla _v {\hat{X}}$ vanishes for all $v \in N_x\varSigma .$

To obtain the required extension ${\tilde{X}}$ on M, we simply use a cutoff function $\rho \in C^\infty (M)$ which vanishes outside a neighborhood $U' \subset U_\delta$ of $\varSigma$ and such that $\rho \equiv 1$ on a smaller neighborhood $U''\subset \!\subset U'$ of $\varSigma ,$ and let ${\tilde{X}}=\rho {\hat{X}}.$$\square$