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Register a new userSplitting of 3-Manifolds and Rigidity of Area-Minimising Surfaces
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In this paper we prove an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. This result is a variant of a comparison theorem of Heintze-Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient manifold are weaker but the assumptions on the surface are considerably more restrictive. We then use our comparison theorem to provide a unified proof of various splitting theorems for 3-manifolds with lower bounds on the scalar curvature.
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By work of Uhlenbeck, the largest principal curvature of any least area fiber of a hyperbolic $3$-manifold fibering over the circle is bounded below by one. We give a short argument to show that, along certain families of fibered hyperbolic $3$-manifolds, there is a uniform lower bound for the maximum principal curvatures of a least area minimal surface which is greater than one.
Unlike Legendrian submanifolds, the deformation problem of coisotropic submanifolds can be obstructed. Starting from this observation, we single out in the contact setting the special class of integral coisotropic submanifolds as the direct generalization of Legendrian submanifolds for what concerns deformation and moduli theory. Indeed, being integral coisotropic is proved to be a rigid condition, and moreover the integral coisotropic deformation problem is unobstructed with discrete moduli space.
Discrete conformal structure on polyhedral surfaces is a discrete analogue of the conformal structure on smooth surfaces, which includes tangential circle packing, Thurstons circle packing, inversive distance circle packing and vertex scaling as special cases and generalizes them to a very general context. Glickenstein conjectured the rigidity of discrete conformal structures on polyhedral surfaces, which includes Luos conjecture on the rigidity of vertex scaling and Bowers-Stephensons conjecture on the rigidity of inversive distance circle packings on polyhedral surfaces as special cases. We prove Glickensteins conjecture using a variational principle. We further study the deformation of discrete conformal structures on polyhedral surfaces by combinatorial curvature flows. It is proved that the combinatorial Ricci flow for discrete conformal structures, which is a generalization of Chow-Luos combinatorial Ricci flow for circle packings and Luos combinatorial Yamabe flow for vertex scaling, could be extended to exist for all time and the extended combinatorial Ricci flow converges exponentially fast for any initial data if the discrete conformal structure with prescribed combinatorial curvature exists. This confirms another conjecture of Glickenstein on the convergence of the combinatorial Ricci flow and provides an effective algorithm for finding discrete conformal structures with prescribed combinatorial curvatures. The relationship of discrete conformal structures on polyhedral surfaces and 3-dimensional hyperbolic geometry is also discussed. As a result, we obtain some new convexities of the co-volume functions for some generalized 3-dimensional hyperbolic tetrahedra.
We derive some elliptic differential inequalities from the Weitzenbock formulas for the traceless Ricci tensor of a Kahler manifold with constant scalar curvature and the Bochner tensor of a Kahler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some $L^p$ and $L^infty $ pinching results are established to characterize Kahler-Einstein manifolds among Kahler manifolds with constant scalar curvature, and others are given to characterize complex space forms among Kahler-Einstein manifolds. Finally, these pinching results may be combined to characterize complex space forms among Kahler manifolds with constant scalar curvature.
We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity case the determinant is analyzed through the zeta-regularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics. These results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances.
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