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Maps from Riemannian manifolds into non-degenerate Euclidean cones

Luciano Mari, Marco Rigoli

Abstract

Let $M$ be a connected, non-compact $m$-dimensional Riemannian manifold. In this paper we consider smooth maps $φ: M \to \mathbb{R}^n$ with images inside a non-degenerate cone. Under quite general assumptions on $M$, we provide a lower bound for the width of the cone in terms of the energy and the tension of $φ$ and a metric parameter. As a side product, we recover some well known results concerning harmonic maps, minimal immersions and Kähler submanifolds. In case $φ$ is an isometric immersion, we also show that, if $M$ is sufficiently well-behaved and has non-positive sectional curvature, $φ(M)$ cannot be contained into a non-degenerate cone of $\mathbb{R}^{2m-1}$.

Maps from Riemannian manifolds into non-degenerate Euclidean cones

Abstract

Let be a connected, non-compact -dimensional Riemannian manifold. In this paper we consider smooth maps with images inside a non-degenerate cone. Under quite general assumptions on , we provide a lower bound for the width of the cone in terms of the energy and the tension of and a metric parameter. As a side product, we recover some well known results concerning harmonic maps, minimal immersions and Kähler submanifolds. In case is an isometric immersion, we also show that, if is sufficiently well-behaved and has non-positive sectional curvature, cannot be contained into a non-degenerate cone of .

Paper Structure

This paper contains 10 sections, 8 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Proof of the Theorems
  3. Proof of Theorem \ref{['stimacono']}
  4. Proof of Corollary \ref{['atsuji']}
  5. Proof of Corollary \ref{['propria']}
  6. Proof of Corollary \ref{['inviluppo']}
  7. Proof of Theorem \ref{['stimacono2']}
  8. Proof of Corollary \ref{['sezminzero']}
  9. Proof of Corollary \ref{['kahler']}
  10. Ending remarks

Key Result

Theorem 1.4

Let M be a connected, non-compact $m$-dimensional Riemannian manifold, and let be a map of class $C^2$ such that $|d\varphi(x)|^2>0$ on $M$. Consider the elliptic operator $L=|d\varphi|^{-2}\Delta$, and assume that $M$ is $L$-stochastically complete. Let $\mathcal{C}=\mathcal{C}_{o,v,\theta}$ be a cone with vertex at $o\in\mathbb{R}^n\backslash\varphi(M)$, let $\pi_v$ be the h In case $\varphi$ i

Theorems & Definitions (15)

  • Theorem 1.4
  • Remark 1
  • Remark 2
  • Remark 3: Sharpness of inequality \ref{['principale']}
  • Corollary 1.9
  • Corollary 1.10
  • Remark 4
  • Definition 1.11
  • Corollary 1.12
  • Theorem 1.14
  • ...and 5 more