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Register a new userMean curvature and compactification of surfaces in a negatively curved Cartan-Hadamard manifold
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We state and prove a Chern-Osserman-type inequality in terms of the volume growth for complete surfaces with controlled mean curvature properly immersed in a Cartan-Hadamard manifold $N$ with sectional curvatures bounded from above by a negative quantity $K_{N}leq b<0$
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Let $nge 2$ and $kge 1$ be two integers. Let $M$ be an isometrically immersed closed $n$-submanifold of co-dimension $k$ that is homotopic to a point in a complete manifold $N$, where the sectional curvature of $N$ is no more than $delta<0$. We prove that the total squared mean curvature of $M$ in $N$ and the first non-zero eigenvalue $lambda_1(M)$ of $M$ satisfies $$lambda_1(M)le nleft(delta +frac{1}{operatorname{Vol} M}int_M |H|^2 operatorname{dvol}right).$$ The equality implies that $M$ is minimally immersed in a metric sphere after lifted to the universal cover of $N$. This completely settles an open problem raised by E. Heintze in 1988.
We describe the action of the fundamental group of a closed Finsler surface of negative curvature on the geodesics in the universal covering in terms of a flat symplectic connection and consider the first order deformation theory about a hyperbolic metric. A construction of O.Biquard yields a family of metrics which give nontrivial deformations of the holonomy, extending the representation of the fundamental group from SL(2,R) into the group of Hamiltonian diffeomorphisms of S^1 x R, and producing an infinite-dimensional version of Teichmuller space which contains the classical one.
We state and prove a Chern-Osserman Inequality in terms of the volume growth for minimal surfaces properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity.
Let $C$ be a strictly convex domain in a $3$-dimensional Riemannian manifold with sectional curvature bounded above by a constant and let $Sigma$ be a constant mean curvature surface with free boundary in $C$. We provide a pinching condition on the length of the traceless second fundamental form on $Sigma$ which guarantees that the surface is homeomorphic to either a disk or an annulus. Furthermore, under the same pinching condition, we prove that if $C$ is a geodesic ball of $3$-dimensional space forms, then $Sigma$ is either a spherical cap or a Delaunay surface.
Motivated by a recent groundbreaking work of Ontaneda, we describe a sizable class of closed manifolds such that the product of each manifold in the class with the real line admits a complete metric of bounded negative sectional curvature which is an exponentially warped near one end and has finite volume near the other end.
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