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In this paper, we study minimal Legendrian surfaces Σ immersed in the tangent sphere bundle T1ℝ3. We classify (1) totally geodesic Legendrian surfaces, (2) closed minimal Legendrian surfaces of genus smaller than or equal to 1 and complete minimal Legendrian surfaces with the non-negative Gauss curvature, and (3) complete stable minimal Legendrian surfaces. [ABSTRACT FROM AUTHOR]

Let M be a compact hypersurface with constant mean curvature in . Denote by H and S the mean curvature and the squared norm of the second fundamental form of M, respectively. We verify that there exists a positive constant γ(n) depending only on n such that if ∣H∣ ⩽ γ(n) and , then S ≡ β(n, H) and M is a Clifford torus. Here, . [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a boundary problem for Lagrangian submanifolds, analogous to the problem for free boundary hypersurfaces and capillary hypersurfaces. Then we present several interesting examples of Lagrangian submanifolds satisfying this boundary condition and we prove a Lagrangian version of the Nitsche (or Hopf) type theorem. Some problems are proposed at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we firstly verify that if Mn is an n-dimensional complete self-shrinker with polynomial volume growth in ℝn+1, and if the squared norm of the second fundamental form of M satisfies 0 ⩽ S − 1 ⩽ 1 18 , then S ≡ 1 and M is a round sphere or a cylinder. More generally, let M be a complete λ-hypersurface of codimension one with polynomial volume growth in ℝn+1 with λ ≠ 0. Then we prove that there exists a positive constant γ, such that if |λ| ⩽ γ and the squared norm of the second fundamental form of M satisfies 0 ⩽ S − β λ ⩽ 1 18 , then S ≡ βλ, λ > 0 and M is a cylinder. Here β λ = 1 2 (2 + λ 2 + | λ | λ 2 + 4) . [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of quadratic Riemannian curvature functionals on closed smooth manifold M of dimension n ⩾ 3 on the space of Riemannian metrics on M consisting of metrics with unit volume. We study the stability of these functionals at the metric with constant sectional curvature as its critical point. [ABSTRACT FROM AUTHOR]

This article is an attempt to understand harmonic and holomorphic maps between two bounded symmetric domains in special situations. We study foliations associated to a lattice-equivariant harmonic map of small rank from a complex ball to another. The result is related to rigidity of some complex two ball quotients. Some open questions are raised as well. [ABSTRACT FROM AUTHOR]

We investigate rigidity problems for odd-dimensional compact submanifolds. We show that if M ( n ⩾ 5) is an odd-dimensional compact submanifold with parallel mean curvature in S, and if Ric > ( n−2− $\tfrac{1} {n}$)(1+ H) and H < δ, where δ is an explicit positive constant depending only on n, then M is a totally umbilical sphere. Here H is the mean curvature of M. Moreover, we prove that if M ( n ⩾ 5) is an odd-dimensional compact submanifold in the space form F( c) with c ⩾ 0, and if Ric > ( n−2− ɛ)( c+ H), where ɛ is an explicit positive constant depending only on n, then M is homeomorphic to a sphere. [ABSTRACT FROM AUTHOR]