The uniqueness of minimal maps into Cartan-Hadamard manifolds via the squared singular values

TL;DR

The paper addresses the uniqueness of minimal maps into Cartan–Hadamard manifolds by leveraging a convexity framework based on squared singular values $\lambda^2=df"){(\lambda_1^2,\dots,\lambda_m^2)}$ along a geodesic homotopy between two maps. It derives the second variation formula for the graph volume and establishes a confinement principle via majorization that ensures $\lambda^2(t)\in \overline{\mathcal{N}}$ along the homotopy. This leads to a stepwise vanishing of the variation and, ultimately, to $f_0=f_1$ under explicit symmetric convex constraints on $\lambda^2(df_0)$ and $\lambda^2(df_1)$, with concrete corollaries such as product- and sum-based bounds that improve prior results. The approach broadens uniqueness/stability results from codimension-one cases to general nonpositively curved targets and provides practical criteria for when Dirichlet data yield a unique minimal map solution.

Abstract

In this paper, we give a uniqueness theorem for the Dirichlet problem of minimal maps into general Riemannian manifolds with non-positive sectional curvature, improving Theorem 5.2 of Lee-Ooi-Tsui's paper published in J. Geom. Anal.. The proof of this theorem is based on the convexity of several functions in terms of squared singular values along the geodesic homotopy of two given minimal maps.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Proposition 3.4