Mathematics

We show how a rigidity estimate introduced in recent work of Bernand-Mantel, Muratov and Simon can be derived using the harmonic map flow.

We revisit and generalize a recent result of Cederbaum [C2, C3] concerning the rigidity of the Schwarzschild manifold for spin manifolds. This includes the classical black hole uniqueness theorems [BM, GIS, Hw] as well as the more recent uniqueness theorems for pho-ton spheres [C1, CG1, CG2].

Let (Z,?) be a connected Kahler manifold with an holomorphic action of the complex reductive Lie group U C , where U is a compact connected Lie group acting in a hamiltonian fashion. Let G be a closed compatible Lie group of U C and let M be a G -invariant connected submanifold of Z . Let x?�M . If G is a real form of U C , we investigate conditions such that G?�x compact implies U C ?�x is compact as well. The vice-versa is also investigated. We also characterize G -invariant real submanifolds such that the norm square of the gradient map is constant. As an application, we prove a splitting result for real connected submanifolds of (Z,?) generalizing a result proved in \cite{pg}, see also \cite{bg,bs}.

In this paper, we investigate space-like graphs defined over a domain Ω??M n in the Lorentz manifold M n ?R with the metric ?�d s 2 +? , where M n is a complete Riemannian n -manifold with the metric ? , Ω has piecewise smooth boundary, and R denotes the Euclidean 1 -space. We can prove an interesting stability result for translating space-like graphs in M n ?R under a conformal transformation.

In complex Finsler geometry, an open problem is: does there exist a weakly Kähler Finsler metric which is not Kähler? In this paper, we give an affirmative answer to this open problem. More precisely, we construct a family of the weakly Kähler Finsler metrics which are non-Kähler. The examples belong to the unitary invariant complex Randers metrics. Furthermore, a uniformization theorem of the unitary invariant complex Randers metrics with constant holomorphic curvature is proved under the weakly Kähler condition.

We consider a smooth fibration equipped with a flat complex vector bundle and a hypersurface cutting the fibration into two pieces. Our main result is a gluing formula relating the Bismut-Lott analytic torsion form of the whole fibration to that of each piece. This result confirms a conjecture proposed in a conference in Goettingen in 2003. Our approach combines an adiabatic limit along the normal direction of the hypersurface and a Witten type deformation on the flat vector bundle.

In this paper, we compute sub-Riemannian limits of Gaussian curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for a Euclidean C 2 -smooth surface in the Heisenberg group away from characteristic points and signed geodesic curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for Euclidean C 2 -smooth curves on surfaces. We get Gauss-Bonnet theorems associated to two kinds of Schouten-Van Kampen affine connections in the Heisenberg group.

Let M be a complex surface. We show that there is a one-to-one correspondence between torsion-free affine connections on M and Riccati distributions on P(TM) . Furthermore, if M is compact, then this correspondence induces a one-to-one correspondence between affine structures on M and Riccati foliations on P(TM) .

For any subgroup of SL(3,R)⋉ R 3 obtained by adding a translation part to a subgroup of SL(3,R) which is the fundamental group of a finite-volume convex projective surface, we first show that under a natural condition on the translation parts of parabolic elements, the affine action of the group on R 3 has convex domains of discontinuity that are regular in a certain sense, generalizing a result of Mess for globally hyperbolic flat spacetimes. We then classify all these domains and show that the quotient of each of them is an affine manifold foliated by convex surfaces with constant affine Gaussian curvature. The proof is based on a correspondence between the geometry of an affine space endowed with a convex cone and the geometry of a convex tube domain. As an independent result, we show that the moduli space of such groups is a vector bundle over the moduli space of finite-volume convex projective structures, with rank equaling the dimension of the Teichmüller space.

In this paper we study affine hypersurfaces with non-degenerate second fundamental form of arbitrary signature additionally equipped with an almost symplectic structure ? . We prove that if R p ?=0 or ??p ?=0 for some positive integer p then the rank of the shape operator is at most one. The results provide complete classification of affine hypersurfaces with higher order parallel almost symplectic forms and are generalization of recently obtained results for Lorentzian affine hypersurfaces.