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Let be a complete noncompact Riemannian manifold. We consider gradient estimates on positive solutions to the following nonlinear equation in , where , are two real constants and , is a smooth real valued function on and . When is finite and the -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 is bounded from above,...
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....
We prove endpoint estimates for operators given by oscillating spectral multipliers on Riemannian manifolds with -bounded geometry and nonnegative Ricci curvature.
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 -irreducible subspaces.
This is a report on some joint work with Aobing Li on Liouville type theorems for some conformally invariant fully nonlinear equations.
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.
In this paper we study the nodal solutions for scalar curvature type equations with perturbation. The main results concern the existence of such solutions and the exact description of their zero set. From this we deduce, in particular cases, some multiplicity results.
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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