A Pisan workshop in Geometric Analysis

# Gradient bounds for the minimal graph equation on Riemannian manifolds.

speaker: Giulio Colombo (Università degli Studi di Milano)

abstract: In this talk we consider the problem of establishing local and global gradient bounds for positive solutions of the minimal graph equation on complete Riemannian manifolds with Ricci curvature bounded below. We show that any entire positive solution $u$ on a complete manifold with $\Ricc\geq-\kappa2$, $\kappa\in[0,+\infty)$, satisfies the bound $\sqrt{1+Du2} \leq e{\kappa u}$, hence we deduce a Liouville theorem for the minimal graph equation on complete manifolds with $\Ricc \geq 0$, also proved by Ding (2019) by different means. On manifolds with $\Ricc \geq 0$ and sectional curvature ($\Sect$) bounded from below by a negative constant, we also show that entire solutions with negative part of linear growth have globally bounded gradient, improving on recent results by Rosenberg, Schulze, Spruck (2013) and Ding, Jost, Xin (2016). As a consequence, we prove that a complete manifold $M$ with $\Sect\geq0$ supporting a nonconstant minimal graphic function of linear growth splits isometrically as a Riemannian product $M = \RR\times M0$. We also discuss different conditions, weaker than $\Sect\geq0$, still ensuring splitting of tangent cones at infinity for complete manifolds with $\Ricc\geq0$ supporting nonconstant minimal graphic functions with bounded gradient. The results presented here are based on joint works with M. Magliaro, L. Mari, M. Rigoli and E. S. Gama.

timetable:
Thu 14 Oct, 10:30 - 12:00, Aula Dini
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