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Entropy Stability for products of negatively curved symmetric spaces

Hyun Chul Jang

Abstract

Let $(M,g_0)$ be a closed oriented $n$-manifold that is locally isometric to a product $(X^{n_1}_1,g_1)\times\cdots (X_k^{n_k},g_k)$, where each $n_i\ge 3,$ and each factor $(X_i^{n_i},g_i)$ is a negatively curved symmetric space. We study the stability of minimal entropy rigidity for such manifolds. Specifically, we consider whether an entropy-minimizing sequence $(M,g_i)$ converges to the model space in the measured Gromov-Hausdorff sense after removing negligible subsets. Previously, Song [Son23] established this type of stability for negatively curved symmetric spaces, where both the $n$-volume of the removed subsets and the $(n-1)$-volume of their boundaries converge to zero. We construct a counterexample demonstrating that this stronger stability notion does not generally hold in the product case; in particular, the condition that the $(n-1)$-volume of the boundary of removed subsets converges to zero cannot be imposed. Nonetheless, we prove that an entropy-minimizing sequence $(M,g_i)$ converges to the model space after removing subsets whose $n$-volume converges to zero in the measured Gromov-Hausdorff topology. This result provides a weaker form of stability compared to the negatively symmetric case. A key ingredient in establishing this stability is our proof of the intrinsic uniqueness of the spherical Plateau solution for products of negatively curved symmetric spaces, which is of independent interest.

Entropy Stability for products of negatively curved symmetric spaces

Abstract

Let be a closed oriented -manifold that is locally isometric to a product , where each and each factor is a negatively curved symmetric space. We study the stability of minimal entropy rigidity for such manifolds. Specifically, we consider whether an entropy-minimizing sequence converges to the model space in the measured Gromov-Hausdorff sense after removing negligible subsets. Previously, Song [Son23] established this type of stability for negatively curved symmetric spaces, where both the -volume of the removed subsets and the -volume of their boundaries converge to zero. We construct a counterexample demonstrating that this stronger stability notion does not generally hold in the product case; in particular, the condition that the -volume of the boundary of removed subsets converges to zero cannot be imposed. Nonetheless, we prove that an entropy-minimizing sequence converges to the model space after removing subsets whose -volume converges to zero in the measured Gromov-Hausdorff topology. This result provides a weaker form of stability compared to the negatively symmetric case. A key ingredient in establishing this stability is our proof of the intrinsic uniqueness of the spherical Plateau solution for products of negatively curved symmetric spaces, which is of independent interest.

Paper Structure

This paper contains 9 sections, 15 theorems, 106 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Uniqueness of the Spherical Plateau solution for products of negatively curved symmetric spaces
  4. The spherical volume
  5. Intrinsic Uniqueness of the Spherical Plateau Solution
  6. Stability for products of negatively curved symmetric spaces
  7. A counterexample of the volume entropy stability conjecture
  8. The volume entropy converges to the minimum volume entropy
  9. The stability statement fails

Key Result

Theorem 1

Let $(M,g_0)$ be a closed oriented $n$-manifold which is locally isometric to a product $(X_1^{n_1},g_1)\times\cdots (X_k^{n_k},g_k)$ of negatively curved symmetric spaces (where $(X_i,g_i)$ has maximum sectional curvature $-1$ after scaling and dimension at least $3$). Define the metric $g_{\mathrm where Then for any Riemannian metric $g$ on $M$ with $\mathrm{Vol}(M,g)=\mathrm{Vol}(M,g_{\mathrm{

Figures (6)

  • Figure 1: The maps $\varphi_i$, $f_i$, and $\Psi$
  • Figure 2: If the turning angle is large, then there exists a shorter path.
  • Figure 3: Shorter path when the angular coordinate is not fixed
  • Figure 4: The region $R_c$ and the comparison of the volume growth of $B_{g_0}(0,\rho)$ and $B_{g_i}(0,\rho)$
  • Figure 5: There exists a point $x\in V(p,\epsilon)$ such that $\partial Z_i\cap P_x$ has small $\mathscr{H}^1$ measure.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Theorem 1: Minimal entropy rigidity for products of negatively curved symmetric spaces Connell:2003aa
  • Theorem 2: Minimal entropy stability for negatively curved symmetric spaces Song:2023aa
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2.1
  • Definition 2.3: Flat topology
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • ...and 24 more