A general form of the weak maximum principle and some applications

Abstract

Abstract. The aim of this paper is to introduce new forms of the weak and Omori-Yau maximum principles for linear operators, notably for trace type operators, and show their usefulness, for instance, in the context of PDE's and in the theory of hypersurfaces. In the final part of the paper we consider a large class of non-linear operators and we show that our previous results can be appropriately generalized to this case.

Lichnerowicz-type equations on complete manifolds

“…Date: July 28, 2018. 1 and ∆ L = div •L is the vector laplacian. The constants appearing in (1.2) are respectively given by…”

Higher order mean curvature estimates for complete hypersurfaces into horoballs

“…In this paper we consider immersed two-sided hypersurfaces ϕ : M → N satisfying a general version of the weak maximum principle established by Albanese, Alias and Rigoli [1] and such that ϕ(M ) is contained in a horoball of N . We prove higher order mean curvature estimates that are extensions of (3) and (4), see Theorems 2.3 and 4.1.…”

“…Stochastic completeness was shown to be equivalent to the weak maximum principle by Pigola, Rigoli and Setti in [28]. Recall that the weak maximum principle for the Laplacian holds on a Riemannian manifold M if for any u ∈ C 2 (M ), bounded above u * : = sup M u < ∞ there exists a sequence of points x k ∈ M such that (1) u(x k ) → u * and △ u(x k ) ≤ 1/k .…”

Complete minimal submanifolds with nullity in Euclidean space

“…The Omori-Yau maximum principle holds on M m under the assumption on the scalar curvature (see [2] or [3, Theorem 2.4]) or if the immersion f is proper (see [3, Theorem 2.5]).…”