# Steepest descent method on a Riemannian manifold: the convex case

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We construct a fundamental solution for a parabolic equation with drift on a Riemannian manifold of nonpositive curvature. We obtain some estimates for this fundamental solution that depend on the conditions on the drift field.

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As is known, the boundary spectral data of a compact Riemannian manifold with boundary are determined by its dynamical boundary data (the response operator of the wave equation) corresponding to any time interval: the response operator is represented in the form of a series over spectral data....

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Let M be an n-dimensional complete non-compact Riemannian manifold, d ï¿½ = e h( x)d V( x) be the weighted measure and $${\triangle_{\mu}}$$ be the weighted Laplacian. In this article, we prove that when the m-dimensional Bakryï¿½ï¿½mery curvature is bounded from below by Ric m = -( m -...

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The question of the preservation of discreteness of the spectrum of the Laplacian acting in a space of differential forms under the cutting and gluing of manifolds reduces to the same problem for compact solvability of the operator of exterior derivation. Along these lines, we give some...

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We construct a fundamental solution of a parabolic equation with drift on a Riemannian manifold of nonpositive curvature by the perturbation method on the basis of a solution of an equation without drift. We establish conditions for the drift field under which this method is applicable.

- Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold. Maeta, Shun // Annals of Global Analysis & Geometry;Jun2014, Vol. 46 Issue 1, p75
We consider biharmonic maps $$\phi :(M,g)\rightarrow (N,h)$$ from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $$ p $$ satisfies $$ 2\le p <\infty $$ . If for such a $$ p $$ , $$\int _M|\tau (\phi )|^{ p }\,\mathrm{d}v_g<\infty $$...

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We prove uniform semi-classical estimates for the resolvent of the SchrÃ¶dinger operatorh2?g +V(x), 0 < hÂ« 1, at a nontrapping energy levelE>0, whereVis a real-valued non-negative potential and ?g denotes the positive Laplace-Beltrami operator on a non-compact complete Riemannian manifold...

- The behavior of bounded solutions of quasilinear elliptic equations on manifolds. Ivanov, A. B. // Mathematical Notes;Aug2009, Vol. 88 Issue 1/2, p53
We consider the behavior of bounded solutions of quasilinear elliptic equations on a special class of Riemannian manifolds. We obtain sufficient conditions for the convergence of solutions to zero.