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An eternal hypersurface flow arising in centro-affine geometry

Xinjie Jiang, Changzheng Qu, Yun Yang

Abstract

In this paper, the existence and uniqueness for a specific centro-affine invariant hypersurface flow in $R^{n+1}$ are studied, and the corresponding evolutionary processes in both centro-affine and Euclidean settings are explored. It turns out that the flow exhibits similar properties as the standard heat flow. In addition, the long time existence of the flow is investigated, which asserts that the hypersurface governed by the flow converges asymptotically toward an ellipsoid via systematically investigating evolutions of the centro-affine invariants. Furthermore, the classification of the eternal solutions for the flow is provided.

An eternal hypersurface flow arising in centro-affine geometry

Abstract

In this paper, the existence and uniqueness for a specific centro-affine invariant hypersurface flow in are studied, and the corresponding evolutionary processes in both centro-affine and Euclidean settings are explored. It turns out that the flow exhibits similar properties as the standard heat flow. In addition, the long time existence of the flow is investigated, which asserts that the hypersurface governed by the flow converges asymptotically toward an ellipsoid via systematically investigating evolutions of the centro-affine invariants. Furthermore, the classification of the eternal solutions for the flow is provided.
Paper Structure (15 sections, 27 theorems, 120 equations)

This paper contains 15 sections, 27 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Evolution equations of the hypersurface flow
  3. Existence and regularity results
  4. The Euclidean support function
  5. The flow in Euclidean setting
  6. A priori estimates
  7. Some results of the flow in Euclidean geometry
  8. The self-similar solutions
  9. The asymptotic behavior of the flow in Euclidean setting
  10. Estimates on the centro-affine invariants
  11. The estimates for Tchebychev vector field
  12. Estimates for the cubic form
  13. Higher derivatives of the cubic form
  14. Proof of Theorem \ref{['mainthmA']}
  15. The eternal solutions of the flow

Key Result

Theorem 1.1

Let $X_0$ be a smooth, closed, uniformly convex hypersurface in ${\mathbb R}^{n+1}$ which encloses the origin. Then the flow mainflow has a unique smooth, uniformly convex solution $X(\cdot,t)$ in $[0,+\infty)$.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 32 more