Isospectral potentials and conformally equivalent isospectral metrics on spheres,balls and Lie groups

We construct pairs of conformally equivalent isospectral Riemannian metrics ?1g and ?2g on spheres Sⁿ and balls Bⁿ⁺¹ for certain dimensions n, the smallest of which is n=7, and on certain compact simple Lie groups. In the case of Lie groups, the metric g is left-invariant. In the case of spheres and balls, the metric g not the standard metric but may be chosen arbitrarily close to the standard one. For the same manifolds (M, g) we also show that the functions ?₁ and ?₂ are isospectral potentials for the Schrödinger operator ?₂\gD + \gf. To our knowledge, these are the first examples of isospectral potentials and of isospectral conformally equivalent metrics on simply connected closed manifolds.     

Isoscattering deformations for complete manifolds of negative curvature

We construct continuous families of nonisometric metrics on simply connected manifolds of dimension n ≥ 9which have the same scattering phase, the same resolvent resonances, and strictly negative sectional curvatures. This situation contrasts sharply with the case of compact manifolds of negative curvature, where Guillemin/Kazhdan, Min-Oo, and Croke/Sharafutdinov showed that there are no nontrivial isospectral deformations of such metrics.     

Small eigenvalues of geometrically finite manifolds

Let M be a complete geometrically finite manifold of bounded negative curvature, infinite volume, and dimension at least 3.We give both a lower bound for the bottom of the spectrum of M and an upper bound for the number of the small eigenvalues of M. These bounds only depend on the dimension, curvature bounds and the volume of the oneneighborhood of the convex core.     

An upper bound for a Hilbert polynomial on quaternionic Kähler manifolds

In this article we prove an upper bound for a Hilbert polynomial on quaternionic Kähler manifolds of positive scalar curvature. As corollaries we obtain bounds on the quaternionic volume and the degree of the associated twistor space. Moreover, the article contains some details on differential equations of finite type. Part of this article is used in the proof of the main theorem.     

Asymptotic expansions of the zeta-determinants of Dirac Laplacians on a compact manifold with boundary

In this article we discuss the asymptotic expansions of the zeta-determinants of Dirac Laplacians on a compact manifold with boundary when the boundary part is stretched. In [12] the author studied the same question under the assumption of no existence of L² - and extended L² -solutions of Dirac operators when the boundary part is stretched to infinite length. Therefore, the results in this article generalize those in [12]. Using the main results we obtain the formula describing the ratio of two zeta-determinants of Dirac Laplacians with the APS boundary conditions associated with two unitary involutions σ₁ and σ₂ on ker B satisfying Gσ_i = -σ_i G. We also prove the adiabatic decomposition formulas for the zeta-determinants of Dirac Laplacians on a closed manifold when the Dirichlet or the APS boundary condition is imposed on partitioned manifolds, which generalize the results in [10] and [11].     

Kähler structures on complex torus

Consider the complex torus T _C under the natural action of the compact real torus T. In this paper, we study T-invariant Kähler structures ω on T_C. For each ω, we consider the corresponding line bundleL on T _C. Namely, the Chern class ofL is [ω], and L is equipped with a connection ? whose curvature is ω. We construct a canonical T-invariant L ²-structure on the sections ofL,and let H _ω be the square-integrable holomorphic sections ofL.Then the Hilbert space H _ω is a unitary T-representation, and we study the multiplicity of the (l-dimensional) irreducible unitary T-representations in H_ω. We shall see that the multiplicity is controlled by the image of the moment map corresponding to the T-action preserving ω.     

A uniformization theorem for complete Kähler manifolds with positive holomorphic bisectional curvature

In the theory of complex geometry, one of the famous problems is the following conjecture of Greene and Wu [13] and Yau [33]: Suppose M is a complete noncompact Kähler manifold with positive holomorphic bisectional curvature; then M is biholomorphic to ?ⁿ. In this paper we use the Ricci flow evolution equation to study this conjecture and prove the result that if M has bounded and positive curvature such that the L’ norm of the curvature on geodesic ball is small enough, then the conjecture is true. Our result gives an improvement on the results of Mok et al. [21] and Mok [22].     

Hyper-Lagrangian submanifolds of Hyperkähler manifolds and mean curvature flow

In this article, we define a new class of middle dimensional submanifolds of a Hyperkähler manifold which contains the class of complex Lagrangian submanifolds, and show that this larger class is invariant under the mean curvature flow. Along the flow, the complex phase map satisfies the generalized harmonic map heat equation. It is also related to the mean curvature vector via a first order differential equation. Moreover, we proved a result on nonexistence of Type I singularity.     

On the stability of the behavior of random walks on groups

We show that, for random walks on Cayley graphs, the long time behavior of the probability of return after 2n steps is invariant by quasi-isometry.     

Kähler-Einstein metrics and projective embeddings

We prove that the complex projective space equipped with its Fubini-Study metric admits no compact Kähler-Einstein submanifold with nonpositive Einstein constant. In particular, the Calabi-Yau metrics carried by an algebraic K3 surface cannot be realized by projective embeddings.     

Existence and multiplicity of normal geodesics in Lorentzian manifolds

In this paper we shall prove some results pertaining to the existence and multiplicity of normal geodesics joining two given submanifolds of an orthogonal splitting Lorentzian manifold. To this aim, we look for critical points of an unbounded suitable functional by using a Saddle-Point Theorem and the relative category theory.     

Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

The Dirichlet-to-Neumann (DN) map Λ_g: C^∞ (?M) → C^∞(?M) on a compact Riemannian manifold (M, g) with boundary is defined by Λ_gh = ?u/?v¦in{t6M}, where u is the solution to the Dirichlet problem Δu = 0, u¦?M = h and v is the unit normal to the boundary. If g^t = g + t? is a variation of the metric g by a symmetric tensor field ?, then Λ_g ^t = Λ_g + tΛ_? + o(t). We study the question: How do tensor fields ? look like for which Λ_? =0? A partial answer is obtained for a general manifold, and the complete answer is given in the two cases: For the Euclidean metric and in the 2D-case. The latter result is used for proving the deformation boundary rigidity of a simple 2-manifold.     

On the gluing problem for Dirac operators on manifolds with cylindrical ends

Combining elements of the b-calculus and the theory of elliptic boundary value problems, we solve the gluing problem for b-determinants of Dirac type operators on manifolds with cylindrical ends. As a corollary of our proof, we derive a gluing formula for the b-eta invariant and also a relative invariant formula relating the b-spectral invariants on a manifold with cylindrical end to the spectral invariants with the augmented APS boundary condition on the corresponding compact manifold with boundary.     

Sequences of stable bundles over compact complex surfaces

When identified with sequences of irreducible Hermitian-Einstein connections, sequences of stable holomorphic bundles of fixed topological type and bounded degree on a compact complex surface equipped with a Gauduchon metric are shown to have strongly convergent subsequences after blowing-up and pulling-back sufficiently many times.     

Isoperimetric inequality in the Grushin plane

We prove a sharp isoperimetric inequality in the Grushin plane and compute the corresponding isoperimetric sets.     

Covering radii and paving diameters of Alexandrov spaces

Covering radii and paving diameters are defined, and the borderline case when cov_k X = π/2, k = 1,…,n + 1 and pav_k X = π/2, k = 1,…,n + 1 is studied (curv X ≥1, dim X = n).     

Regularity of harmonic maps into a static Lorentzian manifold

We study the regularity of harmonic maps from Riemannian manifold into a static Lorentzian manifold. We show that when the domain manifold is two-dimensional, any weakly harmonic map is smooth. We also show that when dimension n of the domain manifold is greater than two, there exists a weakly harmonic map for the Dirichlet problem which is smooth except for a closed set whose (n − 2)-dimensional Hausdorff measure is zero.     

Some properties of the distance function and a conjecture of De Giorgi

In the article [2] Ennio De Giorgi conjectured that any compact n-dimensional regular submanifold M of ℝ ^n+m ,moving by the gradient of the functional