[Paper Review] Lower Semi-Continuity of the Index in the Visosity Method for Minimal Surfaces

Tristan Rivière|arXiv (Cornell University)|Jul 30, 2018

Geometric Analysis and Curvature Flows7 references4 citations

TL;DR

This paper establishes a Hilbert manifold structure on the space of $W^{3,2}$-immersed, oriented closed surfaces in a submanifold of $\mathbb{R}^Q$, enabling a rigorous analysis of the index of minimal surfaces. Using this structure, it proves the lower semi-continuity of the Morse index under viscous approximation and bubble tree convergence, ensuring stability of the index in the limit as the viscosity parameter vanishes.

ABSTRACT

The goal of the present work is twofold. First we prove the existence of an Hilbert Manifold structure on the space of immersed oriented closed surfaces with three derivatives in $L^2$ in an arbitrary sub-manifold $M^m$ of an euclidian space $R^Q$. Second, using this Hilbert manifold structure, we prove a lower semi continuity property of the index for sequences of conformal immersions, critical points to the viscous approximation of the area satisfying Struwe entropy estimate and bubble tree strongly converging in $W^{1,2}$ to a limiting minimal surface as the viscous parameter is going to zero.

Motivation & Objective

• To establish a Hilbert manifold structure on the space of $W^{3,2}$-immersed, oriented closed surfaces in a submanifold $M^m \subset \mathbb{R}^Q$.
• To analyze the behavior of the Morse index under viscous approximation and bubble tree convergence of conformal immersions.
• To prove lower semi-continuity of the index for sequences of critical points of the viscous area functional converging strongly in $W^{1,2}$ to a minimal surface.
• To construct a quotient space of immersions modulo diffeomorphisms preserving marked points, ensuring smooth structure and invariance under reparametrization.

Proposed method

• Constructs the Hilbert manifold $\mathfrak{M}_b(M^m)$ as a disjoint union of quotients of $W^{3,2}$-immersions by positive $W^{3,2}$-diffeomorphisms preserving marked points.
• Introduces a linear operator $D^*_{\vec{\Phi}}$ mapping tangent vectors to the space of $W^{2,2}$-sections of $((\wedge^{1,0}\Sigma)^{\otimes 2})$, projecting onto the orthogonal complement of holomorphic quadratic forms.
• Uses a slice theorem construction via equivariant smooth maps $\vec{w}_{\vec{\Phi}}$ and $\Psi_{\vec{\Phi}}$ to achieve local trivialization of the quotient space.
• Applies Struwe's entropy estimate and strong $W^{1,2}$ convergence of bubble trees to control the second variation of the area functional in the viscous limit.
• Derives a precise pointwise expression for the second variation $D^2F(\vec{\Phi})(\vec{w},\vec{w})$ in terms of the second fundamental form $\vec{\mathbb{I}}_{\vec{\Phi}}$, the metric $g_{\vec{\Phi}}$, and derivatives of the variation $\vec{w}$.
• Establishes a uniform $L^2$-bound on the second variation via integration over $\Sigma$, leading to the key estimate (A.21) controlling the index in the limit.

Experimental results

Research questions

• RQ1 Can a Hilbert manifold structure be rigorously constructed on the space of $W^{3,2}$-immersed surfaces modulo diffeomorphisms preserving marked points, despite the non-Banach Lie group structure of the diffeomorphism group?
• RQ2 Does the Morse index of critical points of the viscous area functional remain lower semi-continuous under strong $W^{1,2}$ convergence to a minimal surface in the vanishing viscosity limit?
• RQ3 How does the second variation of the area functional behave under viscous approximation, and can it be bounded uniformly in terms of the second fundamental form and variation fields?
• RQ4 To what extent do bubble tree convergence and Struwe's entropy estimate control the index stability in the limit?
• RQ5 Can the quotient space $\mathfrak{M}_b(M^m)$ be endowed with a smooth Hilbert manifold structure compatible with the action of diffeomorphisms?

Key findings

• The space $\mathfrak{M}_b(M^m)$ of $W^{3,2}$-immersed surfaces modulo diffeomorphisms preserving marked points admits a well-defined Hilbert manifold structure.
• The canonical projection $\Pi: \mathrm{Imm}_b(M^m) \to \mathfrak{M}_b(M^m)$ is smooth, enabling differential geometric analysis on the quotient.
• For sequences of conformal immersions critical to the viscous area functional, the Morse index is lower semi-continuous in the limit as the viscosity parameter tends to zero.
• The second variation $D^2F(\vec{\Phi})(\vec{w},\vec{w})$ is bounded from below by a positive definite quadratic form involving $|\vec{\mathbb{I}}_{\vec{\Phi}}|^{2}$ and $|\partial\vec{w}|^2$, ensuring stability of the index.
• The estimate (A.21) shows that $|D^2F(\vec{\Phi})(\vec{v}(\vec{\Phi}),\vec{v}(\vec{\Phi}))| \leq C \int_{\Sigma} (1 + |\vec{\mathbb{I}}_{\vec{\Phi}}|^2) (|\partial\vec{v}|^2 + |\partial^2\vec{v}|^2) \, dvol_{g_{\vec{\Phi}}}$, providing uniform control in the limit.
• The lower semi-continuity of the index follows from the uniform control of the second variation and the strong $W^{1,2}$ convergence of the bubble tree, confirming index stability in the viscous limit.