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Modified mean curvature flow of graphs in Riemannian manifolds

Jocel Faustino Norberto de Oliveira, Jorge Herbert Soares de Lira, Matheus Nunes Soares

TL;DR

This work studies the long-time existence of smooth Killing graphs evolving by a modified mean curvature flow in warped product manifolds \\bar{M} = M \\times_\\varrho \\mathbb{R} with a nonvanishing Killing field X. The flow is recast as a quasilinear parabolic PDE for a graph function u on M, with \\partial_t u = \\mathcal{Q}[u] and W = (\\varrho^{-2} + | abla^M u|^2)^{1/2}, linking the geometric evolution to the variational problem of the constrained area functional \\mathcal{A}_\\sigma[u]. The authors establish robust a priori estimates—height, gradient, and curvature—via Hessian comparison, barrier constructions, and Simons-type identities, under natural curvature and warping assumptions. These estimates yield local existence on geodesic balls and, through an exhaustion argument, a global, smooth, longtime solution for locally Lipschitz initial Killing graphs when the parameter \\sigma satisfies a sharp upper bound. The results extend earlier Euclidean and hyperbolic cases to a general warped-product setting, providing a unified framework for modified mean curvature flow of Killing graphs with strong regularity and existence guarantees.

Abstract

We obtain height, gradient, and curvature a priori estimates for a modified mean curvature flow in Riemannian manifolds endowed with a Killing vector field. As a consequence, we prove the existence of smooth, entire, longtime solutions for this extrinsic flow with smooth initial data.

Modified mean curvature flow of graphs in Riemannian manifolds

TL;DR

This work studies the long-time existence of smooth Killing graphs evolving by a modified mean curvature flow in warped product manifolds \\bar{M} = M \\times_\\varrho \\mathbb{R} with a nonvanishing Killing field X. The flow is recast as a quasilinear parabolic PDE for a graph function u on M, with \\partial_t u = \\mathcal{Q}[u] and W = (\\varrho^{-2} + | abla^M u|^2)^{1/2}, linking the geometric evolution to the variational problem of the constrained area functional \\mathcal{A}_\\sigma[u]. The authors establish robust a priori estimates—height, gradient, and curvature—via Hessian comparison, barrier constructions, and Simons-type identities, under natural curvature and warping assumptions. These estimates yield local existence on geodesic balls and, through an exhaustion argument, a global, smooth, longtime solution for locally Lipschitz initial Killing graphs when the parameter \\sigma satisfies a sharp upper bound. The results extend earlier Euclidean and hyperbolic cases to a general warped-product setting, providing a unified framework for modified mean curvature flow of Killing graphs with strong regularity and existence guarantees.

Abstract

We obtain height, gradient, and curvature a priori estimates for a modified mean curvature flow in Riemannian manifolds endowed with a Killing vector field. As a consequence, we prove the existence of smooth, entire, longtime solutions for this extrinsic flow with smooth initial data.
Paper Structure (9 sections, 13 theorems, 247 equations)

This paper contains 9 sections, 13 theorems, 247 equations.

Table of Contents

  1. Introduction
  2. Variational and non-parametric formulations
  3. Hessian comparison principle and fundamental inequalities
  4. Height estimates
  5. Examples of modified MCF in rotationally symmetric metrics
  6. Barriers and height estimates for the modified mean curvature flow
  7. Gradient bounds
  8. Curvature estimates
  9. Proof of the main existence result

Key Result

Theorem 1

Let $M$ be a $n$-dimensional complete, non-compact oriented Riemannian manifold with a pole $o$ and let $\bar{M}$ be the warped product $M\times_\varrho\mathbb{R}$ for some positive function $\varrho\in C^\infty(M)$. Suppose that the curvature conditions cond-1, cond-3 and cond-4 hold. Given a local

Theorems & Definitions (24)

  • Theorem 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • ...and 14 more