Rigidity of minimal submanifolds in hyperbolic space

We prove that if an n-dimensional complete minimal submanifold M in hyperbolic space has sufficiently small total scalar curvature then M has only one end. We also prove that for such M there exist no nontrivial L ² harmonic 1-forms on M.     

On the curvature of minimal submanifolds in a sphere

S. T. Yau proved inAmer. J. Math. 97 (1975), p. 95, Theorem 15 that if the sectional curvature of ann-dimensional compact minimal submanifold in the (n + p)-dimensional unit sphere is everywhere greater than (p – 1)/(2p – 1), then this minimal submanifold is totally geodesic. In this note we improve this bound for the casep 2 to (3p – 2)/(6p).     

Rigidity of minimal submanifolds with flat normal bundle

Let M ⁿ (n ≥ 3) be an n-dimensional complete immersed $ \frac{{n - 2}} {n} $ \frac{{n - 2}} {n} -super-stable minimal submanifold in an (n + p)-dimensional Euclidean space ℝ ^n+p with flat normal bundle. We prove that if the second fundamental form of M satisfies some decay conditions, then M is an affine plane or a catenoid in some Euclidean subspace.     

Complete minimal submanifolds of compact Lie groups

We give a new method for manufacturing complete minimal submanifolds of compact Lie groups and their homogeneous quotient spaces. For this we make use of harmonic morphisms and basic representation theory of Lie groups. We then employ our method to construct many examples of compact minimal submanifolds of the special unitary groups.     

Some results on the occurrence of compact minimal submanifolds

Varying the situation considered in Myers theorem, we show, via standard index form techniques, that a complete Riemannian manifold which admits a compact minimal submanifold is necessarily compact, provided a suitable curvature object is positive on the average along the geodesies issuing orthogonally from the minimal submanifold. By slightly recasting this result, one establishes the nonexistence of compact minimal submanifolds (in particular, closed geodesies) in complete noncompact manifolds which obey an appropriate curvature condition. A generalization of a result of Tipler concerning the occurrence of zeros of solutions to the scalar Jacobi equation is also obtained.     

Willmore submanifolds in the unit sphere via isoparametric functions

In this paper, we show that both focal submanifolds of each isoparametric hypersurface in the sphere with six distinct principal curvatures are Willmore, hence all focal submanifolds of isoparametric hypersurfaces in the sphere are Willmore.     

Proper biharmonic submanifolds in a sphere

In this paper, we obtain a constraint of the mean curvature for proper biharmonic submanifolds in a sphere. We give some characterizations of some proper biharmonic submanifolds with parallel mean curvature vector in a sphere. We also construct some new examples of proper biharmonic submanifolds in a sphere.     

Stability of certain minimal submanifolds in compact symmetric spaces of rank two

In [T. Kimura, M.S. Tanaka, Totally geodesic submanifold in compact symmetric spaces of rank two, Tokyo J. Math., in press], the authors obtained the global classification of the maximal totally geodesic submanifolds in compact connected irreducible symmetric spaces of rank two. In this paper, we determine their stability as minimal submanifolds in compact symmetric spaces of rank two.     

First nonzero eigenvalue of a minimal hypersurface in the unit sphere

In this paper, it is shown that the first nonzero eigenvalue λ₁ of the Laplacian operator on a compact immersed minimal hypersurface M in the unit sphere S ⁿ⁺¹ satisfies one of the following $$ (i)\lambda _{1}=n, \quad (ii)\lambda _{1} \leq (1+k_{0})n, \quad (iii)\lambda _{1}\geq n+\frac{n}{2}(nk_{0}-(n-1))$$ where k ₀ is the infimum of the sectional curvatures of M. It is also shown that a compact immersed minimal hypersurface of the unit sphere S ⁿ⁺¹ with λ₁?=?n is either isometric to the unit sphere S ⁿ or else k ₀?<?n ^?1(n?1).     

Rigidity around Poisson submanifolds

We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash–Moser fast convergence method. In the case of one-point submanifolds (fixed points), this implies a stronger version of Conn’s linearization theorem [2], also proving that Conn’s theorem is a manifestation of a rigidity phenomenon; similarly, in the case of arbitrary symplectic leaves, it gives a stronger version of the local normal form theorem [7]. We can also use the rigidity theorem to compute the Poisson moduli space of the sphere in the dual of a compact semisimple Lie algebra [17].     

Flatness of CR submanifolds in a sphere

In Euclidean geometry, for a real submanifold M in E n+a , M is a piece of E n if and only if its second fundamental form is identically zero. In projective geometry, for a complex submanifold M in CP n+a , M is a piece of CP n if and only if its projective second fundamental form is identically zero. In CR geometry, we prove the CR analogue of this fact in this paper.     

4-dimensional minimal CR submanifolds of the sphere S satisfying Chen's equality

In this paper, we classify 4-dimensional minimal CR submanifolds M of the nearly Kähler 6-sphere S⁶(1) which satisfy Chen's equality, i.e. , where δM(p)=τ(p)−infK(p) for every p∈M.     

New examples of Willmore submanifolds in the unit sphere via isoparametric functions

An isometric immersion ${x:M^n\rightarrow S^{n+p}}$ is called Willmore if it is an extremal submanifold of the Willmore functional: ${W(x)=\int\nolimits_{M^n} (S-nH^2)^{\frac{n}{2}}dv}$ , where S is the norm square of the second fundamental form and H is the mean curvature. Examples of Willmore submanifolds in the unit sphere are scarce in the literature. This article gives a series of new examples of Willmore submanifolds in the unit sphere via isoparametric functions of FKM-type.     

Rigidity in the class of orientable compact surfaces of minimal total absolute curvature

Consider an orientable compact surface in three-dimensional Euclidean space with minimum total absolute curvature. If the Gaussian curvature changes sign to finite order and satisfies a nondegeneracy condition along closed asymptotic curves, we show that any other isometric surface differs by at most a Euclidean motion.