The notion the spread of a matrix was first introduced fifty years ago in algebra. In this article, we define the spread of the shape operator by applying the same idea to submanifolds of Riemannian manifolds. We prove that the spread of shape operator is a conformal invariant for any submanifold in a Riemannian manifold. Then, we prove that, for a compact submanifold of a Riemannian manifold, ...

We assign to each nondegenerate Hamiltonian on a closed symplectic manifold a Floer-theoretic quantity called its “boundary depth,” and establish basic results about how the boundary depths of different Hamiltonians are related. As applications, we prove that certain Hamiltonian symplectomorphisms supported in displaceable subsets have infinitely many nontrivial geometrically distinct periodic ...

Let x : M → Sn+p be an n-dimensional submanifold in an (n + p)dimensional unit sphere Sn+p, x : M → Sn+p is called a Willmore submanifold if it is an extremal submanifold to the following Willmore functional: ∫ M (S − nH) 2 dv, where S = ∑ α,i,j (hij) 2 is the square of the length of the second fundamental form, H is the mean curvature of M . In [13], author proved an integral inequality of Sim...

Let Mn be a Riemannian n-manifold. Denote by S(p) and Ric(p) the Ricci tensor and the maximum Ricci curvature on Mn, respectively. In this paper we prove that every C-totally real submanifolds of a Sasakian space form M̄2m+1(c) satisfies S ≤ ( (n−1)(c+3) 4 + n 2 4 H2)g, where H2 and g are the square mean curvature function and metric tensor on Mn, respectively. The equality holds identically if ...

A contact-stationary Legendrian submanifold of S2n+1 is a Legendrian submanifold whose volume is stationary under contact deformations. The simplest contactstationary Legendrian submanifold (actually minimal Legendrian) is the real, equatorial n-sphere S0. This paper develops a method for constructing contact-stationary (but not minimal) Legendrian submanifolds of S2n+1 by gluing together confi...

This is the first article of the two papers in which we investigate the holomorphic and formal flattening problem for a codimension two real submanifold in C with n ≥ 3 near a non-degenerate CR singular point. The problem is motivated from the study of the complex Plateau problem that seeks for the Levi-flat hypersurface bounded by a given real submanifold and is motivated by the classical comp...

This is the second article of the two papers, in which we investigate the holomorphic and formal flattening problem of a non-degenerate CR singular point of a codimension two real submanifold in C with n ≥ 3. The problem is motivated from the study of the complex Plateau problem that looks for the Levi-flat hypersurface bounded by a given real submanifold and by the classical complex analysis p...