Mathematics

We give a complete classification of all κ -noncollapsed, complete ancient solutions to the Kähler Ricci flow with nonnegative bisectional curvature.

In the context of Bartnik's splitting conjecture, Galloway \cite{G} conjectured that for any globally hyperbolic, spatially compact and future timelike geodesically complete manifold (M,g) satisfying the strong energy condition, the future boundary of (M,g) consists of one point. We show that whereas this statement is true in dimension 2 due to a result by Ehrlich and Galloway \cite{EG}, it fails in every higher dimension, and that there are plenty of counterexamples, more precisely: On a manifold of dimension ?? , {\em every} globally hyperbolic conformal class contains future complete metrics satisfying the strong energy condition.

We connect two a priori unrelated topics, theory of geodesically equivalent metrics in differential geometry, and theory of compatible infinite dimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is (n+1)(n+2)/2 dimensional; we describe it completely and show that it is maximal. Another has dimension ≤n+2 and is, in a certain sense, polynomial. We show that a nontrivial polynomial family of compatible Poisson structures of dimension n+2 is unique and comes from a pair of geodesically equivalent metrics. In addition, we generalise a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.

The purpose of this article is to study the strict convexity of the Mabuchi functional along a C 1,1 -geodesic, with the aid of the ϵ -geodesics. We proved the L 2 -convergence of the fiberwise volume element of the ϵ -geodesic. Moreover, the geodesic is proved to be uniformly fiberwise non-degenerate if the Mabuchi functional is ϵ -affine.

We provide a lower value for the volume of a unit vector field tangent to an antipodally Euclidean sphere S 2 depending on the length of an ellipse determined by the indexes of its singularities.

Let M be a closed hyperbolic 3-manifold that admits no infinitesimal conformally-flat deformations. Examples of such manifolds were constructed by Kapovich. Then if g is a Riemannian metric on M with scalar curvature greater than or equal to ?? , we find lower bounds for the areas of stable immersed minimal surfaces Σ in M . Our bounds improve the closer Σ is to being homotopic to a totally geodesic surface in the hyperbolic metric. We also consider a functional introduced by Calegari-Marques-Neves that is defined by an asymptotic count of minimal surfaces in (M,g) . We show this functional to be uniquely maximized, over all metrics of scalar curvature greater than or equal to ?? , by the hyperbolic metric. Our proofs use the Ricci flow with surgery.

In this paper, we extend Gary R. Lawlor's original examples of area-minimizing cones over nonoriented manifolds\cite{lawlor1991sufficient}. From the point view of submanifolds themselves, by considering real nonoriented Grassmannians, complex Grassmannians, quaternion Grassmannians and complex, quaternion projective spaces and Cayley projective plane as the Hermitian orthogonal projection operators uniformly, we prove that there exists a family of opposite cones associated with them. These cones are shown area-minimizing by Lawlor's Curvature Criterion, it also can be seen as direct proofs for these cones being area-minimizing under the perspective of isolated orbits of adjoint action and the perspective of symmetric spaces (\cite{kerckhove1994isolated},\cite{hirohashi2000area},\cite{kanno2002area},\cite{ohno2015area}). Additionally, for the left oriented real Grassmannian manifolds-the double cover of the nonoriented cases, by embedding them in the exterior vector spaces as unit simple vectors, we prove these cones are area-minimizing except one.

The research on minimal cones over products of spheres is a very important problem with a long history, many people have contributed to this issue. It was given a complete answer in \cite{lawlor1991sufficient} by Gary R. Lawlor where he developed a general method for proving that a cone is area-minimizing, the so-called Curvature Criterion. In this paper, we generalize the case of products of spheres to the case of products of embedded Grassmannian manifolds into spheres where they are similar in many ways, and we give a class of area-minimizing cones associated with them by using Curvature Criterion.

We show that a sequence of n-dimensional spaces diffeomorphic to closed aspherical manifolds with non-elementary hyperbolic fundamental groups cannot converge in the Gromov-Hausdorff sense to a lower dimensional compact space under a lower curvature bound.

In this paper, we study the flow of closed, starshaped hypersurfaces in R n+1 with speed r α σ 1/2 2 , where σ 1/2 2 is the normalized square root of the scalar curvature, α≥2, and r is the distance from points on the hypersurface to the origin. We prove that the flow exists for all time and the starshapedness is preserved. Moreover, after normalization, we show that the flow converges exponentially fast to a sphere centered at origin. When α<2, a counterexample is given for the above convergence.