Mathematics

We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with umbilic boundary provided the dimension of the manifold is n>7 and that the Weyl tensor W(x) is not vanishing on the boundary of M.

In this paper we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space R^3 to the case of helicoidal surfaces in the Bianchi-Cartan-Vranceanu (BCV) spaces, i.e. in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension 4 or 6 , except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mean curvature, including the minimal ones.

In the study of moduli spaces defined by the anti-self-dual (ASD) Yang-Mills equations on SU(2) or SO(3) bundles over closed oriented Riemannian 4-manifolds M , the bubble tree compactification was defined in [T88], [F95], [C02], [C10], [F14] and [F15]. The smooth orbifold structure of the bubble tree compactification away from the trivial stratum is defined in [C02] and [C10]. In this paper, we apply the technique to 4-orbifolds of the form M/ Z α satisfying Condition ??? to get a bubble tree compactification of the instanton moduli space on this particular kind of 4-orbifolds.

Given a constant k>1 , let Z be the family of round spheres of radius artanh( k −1 ) in the hyperbolic space H 3 , so that any sphere in Z has mean curvature k . We prove a crucial nondegeneracy result involving the manifold Z . As an application, we provide sufficient conditions on a prescribed function ϕ on H 3 , which ensure the existence of a C 1 -curve, parametrized by ε≈0 , of embedded spheres in H 3 having mean curvature k+εϕ at each point.

We develop a bubble-compactness theory for embedded CMC hypersurfaces with bounded index and area inside closed Riemannian manifolds in low dimensions. In particular we show that convergence always occurs with multiplicity one, which implies that the minimal blow-ups (bubbles) are all catenoids. We also provide bounds on the area of separating CMC surfaces of bounded (Morse) index and use this, together with the previous results, to bound their genus.

Our purpose is to show the existence of a Calabi-Yau structure on the punctured cotangent bundle T ??0 ( P 2 O) of the Cayley projective plane P 2 O and to construct a Bargmann type transformation from a space of holomorphic functions on T ??0 ( P 2 O) to L 2 -space on P 2 O . The space of holomorphic functions corresponds to the Fock space in the case of the original Bargmann transformation. A Kähler structure on T ??0 ( P 2 O) was shown by identifying it with a quadrics in the complex space C 27 ?�{0} and the natural symplectic form of the cotangent bundle T ??0 ( P 2 O) is expressed as a Kähler form. Our method to construct the transformation is the pairing of polarizations, one is the natural Lagrangian foliation given by the projection map q: T ??0 ( P 2 O)??P 2 O and the polarization given by the Kähler structure. The transformation gives a quantization of the geodesic flow in terms of one parameter group of elliptic Fourier integral operators whose canonical relations are defined by the graph of the geodesic flow action at each time. It turn out that for the Cayley projective plane the results are not same with other cases of the original Bargmann transformation for Euclidean space, spheres and other projective spaces.

We prove the central limit theorem of random variables induced by distances to Brownian paths and Green functions on the universal cover of Riemannian manifolds of finite volume with pinched negative curvature. We further provide some ergodic properties of Brownian motions and an application of the central limit theorem to the dynamics of geodesic flows in pinched negative curvature.

In this paper, we investigate the Chern number inequalities on 4 -dimensional Kähler manifolds admitting the deformed Hermitian-Yang-Mills metrics under the assumption θ ^ ∈(π,2π) .

We prove that the Lie algebra S(P(E,M)) of symbols of linear operators acting on smooth sections of a vector bundle E→M, characterizes it. To obtain this, we assume that S(P(E,M)) is seen as C ∞ (M)− module and that the vector bundle is of rank n>1. We improve this result for the Lie algebra S 1 (P(E,M)) of symbols of first-order linear operators. We obtain a Lie algebraic characterization of vector bundles with S 1 (P(E,M)) without the hypothesis of being seen as a C ∞ (M)− module.

In this paper, we consider generalized Yamabe solitons which include many notions, such as Yamabe solitons, almost Yamabe solitons, h-almost Yamabe solitons, gradient k-Yamabe solitons and conformal gradient solitons. We completely classify the generalized Yamabe solitons on hypersurfaces in Euclidean spaces arisen from the position vector field.