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Harmonic projections in negative curvature II: large convex sets

Ognjen Tošić

TL;DR

The paper extends the harmonic map theory in pinched Hadamard manifolds by weakening the quasi-isometry condition to a non-collapsing framework and its sublinear-parameter variant, ω-weakly non-collapsing. It develops a generalized interior estimate that controls how far a harmonic map can deviate from a given Lipschitz map on large scales, enabling the construction of harmonic maps at finite distance from non-collapsing maps and from nearest-point projections to large admissible convex sets. A key technical toolkit includes Cheng-type gradient bounds, nonlinear Schauder estimates, cone-deformation lemmas, and boundary-analysis arguments in hyperbolic geometry. The results apply to convex hulls in hyperbolic spaces and to projections onto admissible convex sets, yielding harmonic maps near convex hulls of regular boundary data and closing gaps in the Schoen conjecture for broader geometric contexts. Together, these contributions broaden the scope of harmonic map existence results in negatively curved spaces and illuminate the geometry of convex sets at infinity.

Abstract

An important result in the theory of harmonic maps is due to Benoist--Hulin: given a quasi-isometry $f:X\to Y$ between pinched Hadamard manifolds, there exists a unique harmonic map at a finite distance from $f$. Here we show existence of harmonic maps under a weaker condition on $f$, that we call non-collapsing -- we require that the following two conditions hold uniformly in $x\in X$: (1) average distance from $f(x)$ to $f(y)$ for $y$ on the sphere of radius $R$ centered at $x$ grows linearly with $R$ (2) the pre-image under $f$ of small cones with apex $f(x)$ have low harmonic measures on spheres centered at $x$. Using these ideas, we also continue the previous work of the author on existence of harmonic maps that are at a finite distance from projections to certain convex sets. We show this existence in a pinched negative curvature setting, when the convex set is large enough. For hyperbolic spaces, this includes the convex hulls of open sets in the sphere at infinity with sufficiently regular boundary.

Harmonic projections in negative curvature II: large convex sets

TL;DR

The paper extends the harmonic map theory in pinched Hadamard manifolds by weakening the quasi-isometry condition to a non-collapsing framework and its sublinear-parameter variant, ω-weakly non-collapsing. It develops a generalized interior estimate that controls how far a harmonic map can deviate from a given Lipschitz map on large scales, enabling the construction of harmonic maps at finite distance from non-collapsing maps and from nearest-point projections to large admissible convex sets. A key technical toolkit includes Cheng-type gradient bounds, nonlinear Schauder estimates, cone-deformation lemmas, and boundary-analysis arguments in hyperbolic geometry. The results apply to convex hulls in hyperbolic spaces and to projections onto admissible convex sets, yielding harmonic maps near convex hulls of regular boundary data and closing gaps in the Schoen conjecture for broader geometric contexts. Together, these contributions broaden the scope of harmonic map existence results in negatively curved spaces and illuminate the geometry of convex sets at infinity.

Abstract

An important result in the theory of harmonic maps is due to Benoist--Hulin: given a quasi-isometry between pinched Hadamard manifolds, there exists a unique harmonic map at a finite distance from . Here we show existence of harmonic maps under a weaker condition on , that we call non-collapsing -- we require that the following two conditions hold uniformly in : (1) average distance from to for on the sphere of radius centered at grows linearly with (2) the pre-image under of small cones with apex have low harmonic measures on spheres centered at . Using these ideas, we also continue the previous work of the author on existence of harmonic maps that are at a finite distance from projections to certain convex sets. We show this existence in a pinched negative curvature setting, when the convex set is large enough. For hyperbolic spaces, this includes the convex hulls of open sets in the sphere at infinity with sufficiently regular boundary.
Paper Structure (21 sections, 14 theorems, 91 equations, 1 figure)

This paper contains 21 sections, 14 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. More precise results
  3. Lipschitz quasi-isometries are non-collapsing
  4. Organization and a brief outline
  5. Acknowledgements
  6. Preliminaries on the geometric analysis of harmonic maps
  7. Deforming to smooth maps
  8. Moving the apex of a cone
  9. Generalized interior estimate
  10. Proof of Lemma \ref{['lm:fundamental-inequality']}
  11. Proof of (\ref{['eq:visual-angle-bound']}).
  12. Proof of (\ref{['eq:int-deficiency']}).
  13. Weakly non-collapsing maps
  14. Proof of Proposition \ref{['prop:limiting']}
  15. Proof of Theorem \ref{['thm:main-qia-omega']}
  16. ...and 6 more sections

Key Result

Theorem 1

For any non-collapsing Lipschitz map $f:X\to Y$ between pinched Hadamard manifolds, there exists a harmonic map $h:X\to Y$ such that $\sup\mathrm{dist}(h, f)<\infty$.

Figures (1)

  • Figure 1: Setup of the proof of Lemma \ref{['lm:fundamental-inequality']}.

Theorems & Definitions (37)

  • Definition 1.1
  • Theorem 1
  • Definition 1.2
  • Theorem 2
  • Theorem 3
  • Definition 1.3
  • Theorem 4
  • Remark 1.4
  • Definition 1.5
  • Theorem 5: Generalized interior estimate
  • ...and 27 more