Bilinear restriction estimates for surfaces with curvatures of different signs

Recently, the sharp -bilinear (adjoint) restriction estimates for the cone and the paraboloid were established by Wolff and Tao, respectively. Their results rely on the fact that for the cone and the paraboloid, the nonzero principal curvatures have the same sign. We generalize those bilinear restriction estimates to surfaces with curvatures of different signs.

     

Compact Hermitian surfaces of constant antiholomorphic sectional curvatures

Compact Hermitian surfaces of constant antiholomorphic sectional curvatures with respect to the Riemannian curvature tensor and with respect to the Hermitian curvature tensor are considered. It is proved: a compact Hermitian surface of constant antiholomorphic Riemannian sectional curvatures is a self-dual Kaehler surface; a compact Hermitian surface of constant antiholomorphic Hermitian sectional curvatures is either a Kaehler surface of constant (non-zero) holomorphic sectional curvatures or a conformally flat Hermitian surface.

     

Marcinkiewicz estimates for solutions of some elliptic problems with nonregular data

The method introduced by Ennio De Giorgi and Guido Stampacchia for the study of the regularity (L ^p , Marcinkiewicz or C ^0,α ) of the weak solutions of Dirichlet problems hinges on the handle of inequalities concerning the integral of on the subsets where |u(x)| is greater than k. In this framework, here we give a contribution with the study of the Marcinkiewicz regularity of the gradient of infinite energy solutions of Dirichlet problems with nonregular data. Dedicated to Juan Luis Vazquez for his 60th birthday (“El verano del Patriarca”, see [19]).     

Kawamata-Viehweg vanishing fails for log del Pezzo surfaces in characteristic 3

We construct a klt del Pezzo surface in characteristic three violating the Kawamata-Viehweg vanishing theorem. As a consequence we show that there exists a Kawamata log terminal threefold singularity which is not Cohen-Macaulay in characteristic three.     

The Codazzi equation for surfaces

In this paper we study the classical Codazzi equation in space forms from an abstract point of view, and use it as an analytic tool to derive global results for surfaces in different ambient spaces. In particular, we study the existence of holomorphic quadratic differentials, the uniqueness of immersed spheres in geometric problems, height estimates, and the geometry and uniqueness of complete or properly embedded Weingarten surfaces.     

Gaussian estimates for elliptic operators with unbounded drift

We consider a strictly elliptic operator

     

Spectral radius, index estimates for Schrödinger operators and geometric applications

In this paper we study the existence of a first zero and the oscillatory behavior of solutions of the ordinary differential equation ^′(vz^′)+Avz=0, where A, v are functions arising from geometry. In particular, we introduce a new technique to estimate the distance between two consecutive zeros. These results are applied in the setting of complete Riemannian manifolds: in particular, we prove index bounds for certain Schrödinger operators, and an estimate of the growth of the spectral radius of the Laplacian outside compact sets when the volume growth is faster than exponential. Applications to the geometry of complete minimal hypersurfaces of Euclidean space, to minimal surfaces and to the Yamabe problem are discussed.     

Affine surfaces with constant affine curvatures

In this paper, we study affine non-degenerate Blaschke immersions from a surface M in ³. We will assume that M has constant affine curvature and constant affine mean curvature, i.e. both the determinant and the trace of the shape operator are constant. Clearly, affine spheres satisfy both these conditions. In this paper, we completely classify the affine surfaces with constant affine curvature and constant affine mean curvature, which are not affine spheres.Research Assistant of the National Fund for Scientific Research (Belgium).     

Edgeworth expansions for spectral mean estimates with applications to Whittle estimates

We prove that the distributions of spectral mean estimates from linear processes admit Edgeworth expansions. As a consequence, Edgeworth expansions are valid for Whittle estimates.     

On surfaces with zero vanishing cycles

We show that using an idea from a paper by Van de Ven one may obtain a simple proof of Zak’s classification of smooth projective surfaces with zero vanishing cycles. This method of proof allows one to extend Zak’s theorem to the case of finite characteristic.     

On some examples of obstructed irregular surfaces

We determine the base space of the Kuranishi family of some complete intersections in the product of an abelian variety and a projective space.As a consequence,we obtain new examples of obstructed irregular surfaces with ample canonical bundle and maximal Albanese dimension.     

A finite volume method on general surfaces and its error estimates

In this paper, we study a finite volume method and its error estimates for the numerical solution of some model second order elliptic partial differential equations defined on a smooth surface. The discretization is defined via a surface mesh consisting of piecewise planar triangles and piecewise polygons. The optimal error estimates of the approximate solution are proved in both the H¹ and L² norms which are of first order and second order respectively under mesh regularity assumptions. Some numerical tests are also carried out to experimentally verify our theoretical analysis.     

Gradient estimates for diffusion semigroups with singular coefficients

Uniform gradient estimates are derived for diffusion semigroups, possibly with potential, generated by second order elliptic operators having irregular and unbounded coefficients. We first consider the Rd-case, by using the coupling method. Due to the singularity of the coefficients, the coupling process we construct is not strongly Markovian, so that additional difficulties arise in the study. Then, more generally, we treat the case of a possibly unbounded smooth domain of Rd with Dirichlet boundary conditions. We stress that the resulting estimates are new even in the Rd-case and that the coefficients can be Hölder continuous. Our results also imply a new Liouville theorem for space-time bounded harmonic functions with respect to the underlying diffusion semigroup.     

On some multivariate density estimates and empirical Bayes problems

This paper consider estimates of multidimensional density functions and their derivatives. The asymptotic unbiasedness and the convergence properties of these estimates are established.Some applications to empirical Bayes problems are considered.     

Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients

Auscher, McIntosh and Tchamitchian studied the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on . In particular, in the case when they obtained Gaussian upper bound estimates for the heat kernel without imposing further assumption on the coefficients. We study the fundamental solutions of the systems of second order parabolic equations in the divergence form with bounded, measurable, time-independent coefficients, and extend their results to the systems of parabolic equations.