調和映射熱流的研究

Abstract

在第一章中，我們回顧了調和映射熱流的理論，並整理了 Yunmei Chen 及Wei-Yue Ding 對爆破解的狄利克雷能量下界的證明，該結果被用來證明調和映射熱流有限時間奇異點的存在性。

在第二章中，我們證明了從 S^m 到 S^n 的任何映射，如果其能量密度至多為min(n, m)/2，則該映射僅可能是等距嵌入到赤道的映射、或是在同倫上平凡。此外，我們還建立了一種系統性方法來研究在若干曲率條件下壓縮映射的剛性，包括愛因斯坦流形和正截面曲率鉗制條件。

在第三章中，我們證明了在複射影空間中，滿足一特定曲率支配條件的的封閉極小子流形是線性子空間。我們還證明了在具有正全純截面曲率的凱勒四維流形中，每個滿足相同曲率支配條件的極小封閉曲面 ι : Σ^2 → X 是凱勒子流形的。 當母空間是代數曲面時，該浸入映射是某單一正常交叉有理代數曲線的正規化映射。

In Chapter 1, we review the theory of harmonic map heat flow and reformulate the proof of the lower bound for the Dirichlet energy of blow-up solutions to harmonic map heat flows established by Yunmei Chen and Wei-Yue Ding. This result is crucial in proving the existence of finite-time singularities in harmonic map heat flow.

In Chapter 2, we generalize Man-Chun Lee and Jingbo Wang’s results in the rigidity of contracting maps. For example, we prove that for any map from S^m to S^n with energy density bounded above by min(n, m)/2, it is either an isometric embedding into an equator or homotopically trivial. In this proof, we establish a systematic method to investigate the rigidity of contracting maps under several curvature conditions, including Einstein manifolds and positive sectional curvature pinching conditions.

In Chapter 3, we prove that minimal varieties in complex projective spaces satisfying some curvature dominance conditions are linear subspaces. We also prove that every minimal closed surface ι : Σ2 → X satisfying the same curvature dominance condition in a Kähler 4-manifold of positive holomorphic sectional curvature is sub-Kähler, and the immersion is the normalization map of simple normal crossing rational algebraic curve when the ambient manifold is an algebraic surface.