The Cauchy problem for Schr\"{o}dinger flows into K\"{a}hler manifolds

TL;DR

This paper establishes local well-posedness for the Schrödinger flow from Euclidean space into compact Kähler manifolds, given sufficiently smooth initial data, advancing understanding of geometric PDEs.

Contribution

It proves local existence and uniqueness of solutions for the Schrödinger flow into Kähler manifolds with initial data in high-regularity Sobolev spaces.

Findings

Proves local well-posedness for initial data in H^{s+1} with s ≥ n/2+4.

Establishes existence and uniqueness of solutions.

Provides a foundation for further analysis of Schrödinger flows into complex manifolds.

Abstract

We prove local well-posedness of the Schr\"{o}dinger flow from $R^n$ into a compact K\{"a}hler manifold $N$ with initial data in $H^{s+1}(R^n, N)$ for $s\geq n/2+4$.