This thesis is concerned with the study of qualitative properties of solutions of the minimal surface equation and of a class of prescribed mean curvature equations on complete Riemannian manifolds. We derive global gradient bounds for non-negative solutions of such equations on manifolds satisfying a uniform Ricci lower bound and we obtain Liouville-type theorems and other rigidity results on Riemannian manifolds with non-negative Ricci curvature. The proof of the aforementioned global gradient bounds for non-negative solutions u is based on the application of the maximum principle to an elliptic differential inequality satisfied by a suitable auxiliary function z=f(u,|Du|), in the spirit of Bernstein’s method of a priori estimates for nonlinear PDEs and of Yau’s proof of global gradient bounds for harmonic functions on complete Riemannian manifolds. The particular choice of the auxiliary function z parallels the one in Korevaar’s proof of a priori gradient estimates for the prescribed mean curvature equation in Euclidean space. The rigidity results obtained in the last part of the thesis include a Liouville theorem for positive solutions of the minimal surface equation on complete Riemannian manifolds with non-negative Ricci curvature, a splitting theorem for complete parabolic manifolds of non-negative sectional curvature supporting non-constant solutions with linear growth of the minimal surface equation, and a splitting theorem for domains of complete parabolic manifolds with non-negative Ricci curvature supporting non-constant solutions of overdetermined problems involving the mean curvature operator.

GLOBAL GRADIENT BOUNDS FOR SOLUTIONS OF PRESCRIBED MEAN CURVATURE EQUATIONS ON RIEMANNIAN MANIFOLDS / G. Colombo ; tutor: M. Rigoli ; coordinatore: V. Mastropietro. - : . Dipartimento di Matematica Federigo Enriques, 2021 Feb 26. ((33. ciclo, Anno Accademico 2020. [10.13130/colombo-giulio_phd2021-02-26].

GLOBAL GRADIENT BOUNDS FOR SOLUTIONS OF PRESCRIBED MEAN CURVATURE EQUATIONS ON RIEMANNIAN MANIFOLDS

G. Colombo
2021-02-26

Abstract

This thesis is concerned with the study of qualitative properties of solutions of the minimal surface equation and of a class of prescribed mean curvature equations on complete Riemannian manifolds. We derive global gradient bounds for non-negative solutions of such equations on manifolds satisfying a uniform Ricci lower bound and we obtain Liouville-type theorems and other rigidity results on Riemannian manifolds with non-negative Ricci curvature. The proof of the aforementioned global gradient bounds for non-negative solutions u is based on the application of the maximum principle to an elliptic differential inequality satisfied by a suitable auxiliary function z=f(u,|Du|), in the spirit of Bernstein’s method of a priori estimates for nonlinear PDEs and of Yau’s proof of global gradient bounds for harmonic functions on complete Riemannian manifolds. The particular choice of the auxiliary function z parallels the one in Korevaar’s proof of a priori gradient estimates for the prescribed mean curvature equation in Euclidean space. The rigidity results obtained in the last part of the thesis include a Liouville theorem for positive solutions of the minimal surface equation on complete Riemannian manifolds with non-negative Ricci curvature, a splitting theorem for complete parabolic manifolds of non-negative sectional curvature supporting non-constant solutions with linear growth of the minimal surface equation, and a splitting theorem for domains of complete parabolic manifolds with non-negative Ricci curvature supporting non-constant solutions of overdetermined problems involving the mean curvature operator.
RIGOLI, MARCO
MASTROPIETRO, VIERI
Riemannian manifold; Ricci lower bound; minimal surface equation; mean curvature operator; gradient bound; Liouville theorem; splitting theorem
Settore MAT/03 - Geometria
GLOBAL GRADIENT BOUNDS FOR SOLUTIONS OF PRESCRIBED MEAN CURVATURE EQUATIONS ON RIEMANNIAN MANIFOLDS / G. Colombo ; tutor: M. Rigoli ; coordinatore: V. Mastropietro. - : . Dipartimento di Matematica Federigo Enriques, 2021 Feb 26. ((33. ciclo, Anno Accademico 2020. [10.13130/colombo-giulio_phd2021-02-26].
Doctoral Thesis
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/813095
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