[Paper Review] Closed Mean Curvature Self-Shrinking Surfaces of Generalized Rotational Type

Peter McGrath|arXiv (Cornell University)|Jul 2, 2015

Geometric Analysis and Curvature Flows11 references3 citations

TL;DR

This paper constructs a new family of closed, embedded mean curvature self-shrinkers in $\mathbb{R}^{2n}$ for each $n \geq 2$, diffeomorphic to $S^{n-1} \times S^{n-1} \times S^1$ and invariant under $SO(n) \times SO(n)$. The construction uses generalized rotational symmetry and solves a geodesic equation in a warped Riemannian metric, proving existence via limiting behavior of profile curves in the orbit space.

ABSTRACT

For each $n\geq 2$ we construct a new closed embedded mean curvature self-shrinking hypersurface in $\mathbb{R}^{2n}$. These self-shrinkers are diffeomorphic to $S^{n-1} imes S^{n-1} imes S^1$ and are $SO(n) imes SO(n)$ invariant. The method is inspired by constructions of Hsiang and these surfaces generalize self-shrinking "tori" diffeomorphic to $S^{n-1} imes S^1$ constructed by Angenent.

Motivation & Objective

• To construct new examples of closed, embedded mean curvature self-shrinkers beyond the sphere and Angenent’s $S^{n-1} \times S^1$ tori.
• To extend the method of rotational symmetry to higher cohomogeneity by using $O(m) \times O(n)$-invariant structures.
• To prove the existence of odd-dimensional closed self-shrinkers with nontrivial topology, specifically $S^{n-1} \times S^{n-1} \times S^1$, for $n \geq 2$.
• To generalize Hsiang’s equivariant differential geometry techniques to the setting of mean curvature flow self-shrinkers.

Proposed method

• Model the self-shrinker as an $O(n) \times O(n)$-invariant hypersurface in $\mathbb{R}^{2n}$, reducing the problem to a profile curve in the orbit space $Q = \{(x,y) \in \mathbb{R}^2 : x,y \geq 0\}$.
• Derive the ODE for the profile curve $\gamma_R$ by requiring it to be a geodesic in the conformally warped metric $g = x^{2(n-1)}y^{2(n-1)}e^{-(x^2+y^2)/2}(dx^2 + dy^2)$.
• Use a one-parameter family of initial conditions $\gamma_R$ with initial angle $\theta = -\pi/4$ and radius $R$, and analyze their behavior as $R \searrow R_*$.
• Apply compactness and smooth dependence on initial conditions to show that the limiting curve $\gamma_{R_*}$ intersects the diagonal $\ell$ (where $x=y$) orthogonally at both endpoints.
• Prove that the reflection of $\gamma_{R_*}$ across $\ell$ forms a smooth, closed, embedded geodesic, which corresponds to a closed self-shrinker via the orbit space correspondence.
• Use contradiction arguments involving the angular derivative $\dot{\theta}$ and metric curvature to rule out non-embedded or non-closed behavior in the limit.

Experimental results

Research questions

• RQ1 Can new closed, embedded self-shrinkers be constructed beyond the sphere and Angenent’s $S^{n-1} \times S^1$ tori in higher dimensions?
• RQ2 Do $O(m) \times O(n)$-invariant self-shrinkers exist for $m,n \geq 2$ with nontrivial topology such as $S^{n-1} \times S^{n-1} \times S^1$?
• RQ3 Is it possible to construct odd-dimensional closed self-shrinkers using generalized rotational symmetry and geodesic flow in a warped metric?
• RQ4 What constraints does the geometry of the orbit space impose on the existence and regularity of such self-shrinkers?
• RQ5 Can the limiting profile curve of a one-parameter family of solutions yield a closed, embedded geodesic that corresponds to a smooth self-shrinker?

Key findings

• For each $n \geq 2$, there exists a closed, embedded mean curvature self-shrinker in $\mathbb{R}^{2n}$ diffeomorphic to $S^{n-1} \times S^{n-1} \times S^1$.
• The self-shrinker is invariant under the $SO(n) \times SO(n)$ action on $\mathbb{R}^{2n}$, generalizing Angenent’s $SO(n)$-invariant tori.
• The profile curve $\gamma_{R_*}$ of the self-shrinker intersects the diagonal $\ell$ (where $x=y$) orthogonally at both endpoints, ensuring smoothness after reflection.
• The limiting curve $\gamma_{R_*}$ is a geodesic in the warped metric $g = x^{2(n-1)}y^{2(n-1)}e^{-(x^2+y^2)/2}(dx^2 + dy^2)$, and its reflection forms a closed embedded geodesic.
• The construction resolves the issue of non-convergence in $C^2$ near the axes by showing that $\gamma_{R_n}$ cannot converge to a geodesic intersecting the $x$- or $y$-axis unless it fails to be a normal graph.
• The proof establishes that any embedded closed geodesic in the orbit space must intersect the diagonal $\ell$ at least twice, confirming topological consistency.