This project concerns an important conjecture which appears to have been solved recently: It concerns minimal surfaces, in particular minimal submanifolds of spheres. It combines PDE and geometry, though the PDE required is not very much. It does involve some basic spectral theory for the Laplacian on a Riemannian manifold. There are many examples known of submanifolds in spheres which are mini...

We prove that a germ of a holomorphic map f between Cn and Cn ′ sending one real-algebraic submanifold M ⊂ Cn into another M ′ ⊂ Cn ′ is algebraic provided M ′ contains no complex-analytic discs and M is generic and minimal. We also propose an algorithm for finding complex-analytic discs in a real submanifold.

The submanifold quantum mechanics was opened by Jensen and Koppe (Ann. Phys. 63 (1971) 586-591) and has been studied for these three decades. This article gives its more algebraic definition and show what is the essential of the submanifold quantum mechanics from an algebraic viewpoint. MCS Codes: 34L40, 35Q40, 81T20, 32C25

The effective dynamic of a quantum particle on the tubular neighbourhood of some closed submanifold of a riemannian manifold can be described by an effective dynamic on the submanifold as the diameter of the tube decreases to zero. It depends whether Dirichletor Neumann conditions are imposed at the boundary of the tube.

We improve Chen-Ricci inequalities for a Lagrangian submanifold Mn of dimension n (n 2) in a 2n -dimensional complex space form M̃2n(4c) of constant holomorphic sectional curvature 4c with a semi-symmetric metric connection and a Legendrian submanifold Mn in a Sasakian space form M̃2n+1(c) of constant φ -sectional curvature c with a semi-symmetric metric connection, respectively.

In this paper, the geometry of invariant submanifolds of a Sasakian manifold are studied. Necessary and sufficient conditions are given on an submanifold of a Sasakian manifold to be invariant submanifold and the invariant case is considered. In this case, we investigate further properties of invariant submanifolds of a Sasakian manifold. M.S.C. 2000: 53C42, 53C15.

Let M0 be a compact, regular q-pseudoconcave compact CR submanifold of a complex manifold G and B a holomorphic vector bundle on G such that dimH ( M0,B ∣∣ M ) = 0 for some fixed r < q. We prove a global homotopy formula with C estimates for r-cohomology of B on arbitrary CR submanifold M close enough to M0.

Let E be a closed set in C, normally with empty interior, and let us consider continuously-differentiable functions on E in the sense of Whitney, in which the differential of the function is automatically included. This means a continuous complex-valued function f(z) on E together with a real-linear mapping dfz : C n → C defined and continuous for z ∈ E, with properties like those of an ordinar...

We propose a unified computational framework for the problem of deformation and rigidity of submanifolds in a homogeneous space under geometric constraint. A notion of 1-rigidity of a submanifold under admissible deformations is introduced. It means every admissible deformation of the submanifold osculates a one parameter family of motions up to 1st order. We implement this idea to the question...