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Gradient estimates for a nonlinear equation Δ f u + c u - α = 0 on complete noncompact manifolds

Jing Zhang, Bingqing Ma (2011)

Communications in Mathematics

Let ( M , g ) be a complete noncompact Riemannian manifold. We consider gradient estimates on positive solutions to the following nonlinear equation Δ f u + c u - α = 0 in M , where α , c are two real constants and α > 0 , f is a smooth real valued function on M and Δ f = Δ - ∇ f ∇ . When N is finite and the N -Bakry-Emery Ricci tensor is bounded from below, we obtain a gradient estimate for positive solutions of the above equation. Moreover, under the assumption that ∞ -Bakry-Emery Ricci tensor is bounded from below and | ∇ f | is bounded from above,...

Gradient estimates for inverse curvature flows in hyperbolic space

Julian Scheuer (2015)

Geometric Flows

We prove gradient estimates for hypersurfaces in the hyperbolic space Hn+1, expanding by negative powers of a certain class of homogeneous curvature functions F. We obtain optimal gradient estimates for hypersurfaces evolving by certain powers p > 1 of F-1 and smooth convergence of the properly rescaled hypersurfaces. In particular, the full convergence result holds for the inverse Gauss curvature flow of surfaces without any further pinching condition besides convexity of the initial hypersurface....

Laplace type operators: Dirichlet problem

Wojciech Kozł (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We investigate Laplace type operators in the Euclidean space. We give a purely algebraic proof of the theorem on existence and uniqueness (in the space of polynomial forms) of the Dirichlet boundary problem for a Laplace type operator and give a method of determining the exact solution to that problem. Moreover, we give a decomposition of the kernel of a Laplace type operator into 𝖲𝖮 ( n ) -irreducible subspaces.

Mass endomorphism, surgery and perturbations

Bernd Ammann, Mattias Dahl, Andreas Hermann, Emmanuel Humbert (2014)

Annales de l’institut Fourier

We prove that the mass endomorphism associated to the Dirac operator on a Riemannian manifold is non-zero for generic Riemannian metrics. The proof involves a study of the mass endomorphism under surgery, its behavior near metrics with harmonic spinors, and analytic perturbation arguments.

Nonlinear elliptic equations involving critical Sobolev exponent on compact Riemannian manifolds in presence of symmetries.

Zindine Djadli (1999)

Revista Matemática Complutense

In this paper, we study a nonlinear elliptic equation with critical exponent, invariant under the action of a subgroup G of the isometry group of a compact Riemannian manifold. We obtain some existence results of positive solutions of this equation, and under some assumptions on G, we show that we can solve this equation for supercritical exponents.

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