Mathematics

We prove that a "cushioned" Hermitian-Einstein-type equation proposed by Demailly in an approach towards a conjecture of Griffiths on the existence of a Griffiths positively curved metric on a Hartshorne ample vector bundle, has an essentially unique solution when the bundle is stable. This result indicates that the proposed approach must be modified in order to attack the aforementioned conjecture of Griffiths.

We study the Minkowski formula of conformal Killing-Yano two-forms in a spacetime of constant curvature. We establish the spacetime Alexandrov theorem with a free boundary.

We extend some convergence results on nonsingular compact Ricci flows in the papers \cite{Ha:1}, \cite{Se:1} and \cite{FZZ:2} to certain infinite volume noncompact cases which are "partially" nonsingular. As an application, for a finite time singularity which is partially type I, it is shown that a blow up limit is a gradient shrinking soliton.

Given a closed connected spin manifold M with non-negative and somewhere positive scalar curvature, we show that the Dirac operator twisted with any flat Hilbert module bundle is invertible.

We establish that for any proper action of a Lie group on a manifold the associated equivariant differentiable cohomology groups with coefficients in modules of C ??-functions vanish in all degrees except than zero. Furthermore let G be a Lie group of CR -automorphisms of a strictly pseudo-convex CR -manifold M . We associate to G a canonical class in the first differential cohomology of G with coefficients in the C ??-functions on M . This class is non-zero if and only if G is essential in the sense that there does not exist a CR -compatible strictly pseudo-convex pseudo-Hermitian structure on M which is preserved by G . We prove that a closed Lie subgroup G of CR -automorphisms acts properly on M if and only if its canonical class vanishes. As a consequence of Schoen's theorem, it follows that for any strictly pseudo-convex CR -manifold M , there exists a compatible strictly pseudo-convex pseudo-Hermitian structure such that the CR-automorphism group for M and the group of pseudo-Hermitian transformations coincide, except for two kinds of spherical CR -manifolds. Similar results hold for conformal Riemannian and Kähler manifolds.

We consider the problem of finding a metric in a given conformal class with prescribed nonpositive scalar curvature and nonpositive boundary mean curvature on a compact manifold with boundary, and establish a necessary and sufficient condition in terms of a conformal invariant that measures the zero set of the target curvatures.

In this paper we study the reduced and unreduced L 2 -cohomology groups of manifolds of bounded geometry and their behavior under uniformly proper lipschitz maps. A \textit{uniformly proper map} is a map such that the diameter of the preimage of a subset depends only on the diameter of the subset. In general, the pullback map along a uniformly proper lipschitz map doesn't induce a morphism in, reduced or not, L 2 -cohomology. Then, our goal is to introduce two contravariant functors between the category of manifolds of bounded geometry and uniformly proper lipschitz maps and the category of complex vector spaces and linear maps. As consequence we obtain that the, reduced or not, L 2 -cohomology is a lipschitz-homotopy invariant. Moreover these functors coincide with the pullback, when the pullback does induce a map between the (un)-reduced L 2 -cohomologies.

In arXiv:1403.7671, Kapovich, Leeb and Porti gave several new characterizations of Anosov representations ??�G , including one where geodesics in the word hyperbolic group ? map to "Morse quasigeodesics" in the associated symmetric space G/K . In analogy with the negative curvature setting, they prove a local-to-global principle for Morse quasigeodesics and describe an algorithm which can verify the Anosov property of a given representation in finite time. However, some parts of their proof involve non-constructive compactness and limiting arguments, so their theorem does not explicitly quantify the size of the local neighborhoods one needs to examine to guarantee global Morse behavior. In this paper, we supplement their work with estimates in the symmetric space to obtain the first explicit criteria for their local-to-global principle. This makes their algorithm for verifying the Anosov property effective. As an application, we demonstrate how to compute explicit perturbation neighborhoods of Anosov representations with two examples.

Using quantization techniques, we show that the δ -invariant of Fujita-Odaka coincides with the optimal exponent in certain Moser-Trudinger type inequality. Consequently we obtain a uniform Yau-Tian-Donaldson theorem for the existence of twisted Kähler-Einstein metrics with arbitrary polarizations. Our approach mainly uses pluripotential theory, which does not involve Cheeger-Colding-Tian theory or the non-Archimedean language. A new computable criterion for the existence of constant scalar curvature Kähler metrics is also given.

We prove an optimal relative integral convergence rate for two expanding gradient Ricci solitons coming out of the same cone. As a consequence, we obtain a unique continuation result at infinity and we prove that a relative entropy for two such self-similar solutions to the Ricci flow is well-defined.