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Local well-posedness of the skew mean curvature flow for large data

Jiaxi Huang, Daniel Tataru

TL;DR

This work proves local well-posedness for the skew mean curvature flow (SMCF) in codimension two with large data in low-regularity Sobolev spaces by reformulating SMCF as a covariant nonlinear Schrödinger system for the second fundamental form $\lambda$ under a dynamic gauge. The authors introduce time-discretization and intrinsic fractional spaces $X^s$, $Y^{s+1}$, and $Z^s$ to handle large data, and develop linearized and parabolic energy estimates to close the argument. Regularized initial data and a passing-to-the-limit strategy yield rough solutions with Hadamard-type well-posedness, including continuous dependence on initial data. The approach avoids nontrapping conditions and low-frequency assumptions of prior work, offering a robust framework for large-data local analysis of geometric quasilinear Schrödinger flows.

Abstract

The skew mean curvature flow is an evolution equation for $d$ dimensional ma\-nifolds embedded in $\mathbb{R}^{d+2}$ (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove large data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $d\geq 2$. This is achieved by introducing several new ideas: (i) a time discretization method to establish the existence of smooth solutions, (ii) constructing the orthonormal frame by a parallel transport method and a lifting criterion, (iii) introducing intrinsic fractional function spaces $X^s\subset H^s$ on a noncompact manifold for any $s>\frac{d}{2}$, such that the $X^s$-norm of the second fundamental form can be propagated well along the quasilinear Schrödinger flow, (iv) deriving a difference equation to prove the uniqueness result for solutions $F\in C^2$, which is independent in the choices of gauge. Our method turns out to be more robust for large data problem.

Local well-posedness of the skew mean curvature flow for large data

TL;DR

This work proves local well-posedness for the skew mean curvature flow (SMCF) in codimension two with large data in low-regularity Sobolev spaces by reformulating SMCF as a covariant nonlinear Schrödinger system for the second fundamental form under a dynamic gauge. The authors introduce time-discretization and intrinsic fractional spaces , , and to handle large data, and develop linearized and parabolic energy estimates to close the argument. Regularized initial data and a passing-to-the-limit strategy yield rough solutions with Hadamard-type well-posedness, including continuous dependence on initial data. The approach avoids nontrapping conditions and low-frequency assumptions of prior work, offering a robust framework for large-data local analysis of geometric quasilinear Schrödinger flows.

Abstract

The skew mean curvature flow is an evolution equation for dimensional ma\-nifolds embedded in (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove large data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension . This is achieved by introducing several new ideas: (i) a time discretization method to establish the existence of smooth solutions, (ii) constructing the orthonormal frame by a parallel transport method and a lifting criterion, (iii) introducing intrinsic fractional function spaces on a noncompact manifold for any , such that the -norm of the second fundamental form can be propagated well along the quasilinear Schrödinger flow, (iv) deriving a difference equation to prove the uniqueness result for solutions , which is independent in the choices of gauge. Our method turns out to be more robust for large data problem.
Paper Structure (37 sections, 39 theorems, 613 equations)

This paper contains 37 sections, 39 theorems, 613 equations.

Table of Contents

  1. Introduction
  2. The (SMCF) equations
  3. Gauge choices for (SMCF)
  4. Scaling and function spaces
  5. The main result
  6. An overview of the paper
  7. The differentiated equations and gauge choices
  8. Notations and the compatibility conditions
  9. The evolutions of metric , connection and the second fundamental form under (SMCF)
  10. The background manifold
  11. The gauge choices
  12. The gauge choices
  13. The modified Schrödinger system
  14. Function spaces and notations
  15. Notations and some properties on manifolds
  16. ...and 22 more sections

Key Result

Proposition 1.1

Assume that $F$ is a solution of Main-Sys with initial data $F(0)=F_0$, then ${F}_\mu(t,x):=\mu^{-1}F(\mu^2 t,\mu x)$ is a solution of Main-Sys with initial data ${F}_\mu(0)=\mu^{-1}F_0(\mu x)$.

Theorems & Definitions (99)

  • Proposition 1.1: Scale invariance for (SMCF)
  • Theorem 1.2: Existence of regular solutions
  • Remark 1.2.1
  • Theorem 1.3: Uniqueness of solutions
  • Theorem 1.4: Local well-posedness for rough data
  • Remark 1.4.1
  • Theorem 2.1: Local well-posedness for large data in the good gauge
  • Lemma 3.1
  • proof
  • Theorem 3.2: Theorem 12.1, p.291Ha1982
  • ...and 89 more