Compact Minimal Submanifolds Research Articles

Compact Minimal Submanifolds in a Large Class of Riemannian Manifolds

Through a new technique, we provide uniqueness, rigidity and non-existence results for compact minimal submanifolds of arbitrary dimension in a large class of Riemannian manifolds, which include between others, Riemannian double-twisted and warped products. Moreover, we show that our results can be applied in particular, to space forms and Cartan–Hadamard manifolds, re-obtaining several classic results in a different approach. Interesting applications to Geometric Analysis are also showed.

Index of compact minimal submanifolds of the Berger spheres

The stability and the index of compact minimal submanifolds of the Berger spheres \({\mathbb {S}}^{2n+1}_{\tau },\, 0<\tau \le 1\), are studied. Unlike the case of the standard sphere (\(\tau =1\)), where there are no stable compact minimal submanifolds, the Berger spheres have stable ones if and only if \(\tau ^2\le 1/2\). Moreover, there are no stable compact minimal d-dimensional submanifolds of \({\mathbb {S}}^{2n+1}_\tau \) when \(1 / (d+1) < \tau ^2 \le 1\) and the stable ones are classified for \(\tau ^2=1 / (d+1)\) when the submanifold is embedded. Finally, the compact orientable minimal surfaces of \({\mathbb {S}}^3_{\tau }\) with index one are classified for \(1/3\le \tau ^2\le 1\).

A generalization of Pólya conjecture and Li–Yau inequalities for higher eigenvalues

We verify that if M is a compact minimal submanifold with boundary in a Cartan–Hadamard manifold N, then $$\sum \nolimits _{i=1}^{k}\lambda _i\ge \frac{2n\pi }{e}k^{\frac{n+2}{n}}{{\,\mathrm{vol}\,}}(M)^{-\frac{2}{n}}$$ , where $$\lambda _i$$ is the ith Dirichlet eigenvalue of the Laplacian on M. This provides an evidence for that the generalized Polya conjecture is true. We also prove that if M is a complete and non-compact minimal submanifold in a Cartan–Hadamard manifold N, then $$\gamma _n N_\alpha (V)\le \int _M(V-\alpha )_{-}^{n/2}(x)dx$$ , where $$N_\alpha (V)$$ is the number of eigenvalues( $$\le \alpha $$ ) of the Schrodinger operator $$-\Delta +V$$ on M and $$\gamma _n$$ is an explicit positive constant depending only on n.

A Pinching Theorem for Compact Minimal Submanifolds in Warped Products I×fSm(c)

Let Sm(c) be a Euclidean sphere of curvature c&gt;0 and R be a Euclidean line. We prove a pinching theorem for compact minimal submanifolds immersed in Riemannian warped products of the type I×fSm(c), where f:I→R+ is a smooth positive function on an open interval I of R. This allows us to generalize Chen-Cui’s pinching theorem from Riemannian products Sm(c)×R to Riemannian warped products I×fSm(c).

A DDVV Type Inequality and a Pinching Theorem for Compact Minimal Submanifolds in a Generalized Cylinder $$\mathbb {S}^{n_1}(c)\times \mathbb {R}^{n_2}$$

In this paper, we prove a new DDVV type inequality for submanifolds immersed in a Riemannian product $$\mathbb {M}^{n_1}(c)\times \mathbb {R}^{n_2} (c\ne 0)$$ of a real space form $$\mathbb {M}^{n_1}(c)$$ of curvature c and a Euclidean space $$\mathbb {R}^{n_2}$$ . In addition, we obtain a pinching theorem for compact minimal submanifolds immersed in a generalized cylinder $$\mathbb {S}^{n_1}(c)\times \mathbb {R}^{n_2}$$ of a sphere $$\mathbb {S}^{n_1}(c)$$ with curvature $$c>0$$ and a Euclidean space $$\mathbb {R}^{n_2}$$ , which extends Lu’s and Chen–Cui’s pinching theorems.

Minimal isoparametric submanifolds of [formula omitted] and octonionic eigenmaps

We use the octonionic multiplication ⋅ of S7 to associate, to each unit normal section η of a submanifold M of S7, an octonionic Gauss map γη:M→S6, γη(x)=x−1⋅η(x), x∈M, where S6 is the unit sphere of T1S7, 1 is the neutral element of ⋅ in S7. Denoting by N(M) the vector bundle of normal sections of M and by E(M) the vector bundle of sections of the vector bundle of endomorphisms of TM eqquiped with the Hilbert–Schmidt metric and defining the bundle homomorphism B:N(M)→E(M) by B(η)=Sη, where Sη is the second fundamental form of M determined by η, we prove that if M is a minimal submanifold of S7 and η∈N(M) is unitary and parallel on the normal connection, then γη is harmonic if and only if η is an eigenvector of B⁎B:N(M)→N(M), where B⁎ is the adjoint of B. If M is an isoparametric compact minimal submanifold of codimension k of S7 then B⁎B has constant non negative eigenvalues 0≤σ1≤⋯≤σk and the associated eigenvectors η1,⋯,ηk form an orthonormal basis of N(M), parallel on the normal connection, such that each γηj is an eigenmap of M with eigenvalue 7−k+σj. Moreover, σj=‖Sηj‖2, 1≤j≤k.

Mean curvature flows in manifolds of special holonomy

We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant--Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean curvature flow. The proof relies on intricate interconnections of the Ricci flatness of the ambient space and the extrinsic geometry of the calibrated submanifolds.

Some results on Finsler submanifolds

In this paper we study the submanifold theory in terms of Chern connection. We introduce the notions of the second fundamental form and mean curvature for Finsler submanifolds, and establish the fundamental equations by means of moving frame for the hypersurface case. We provide the estimation of image radius for compact submanifold, and prove that there exists no compact minimal submanifold in any complete noncompact and simply connected Finsler manifold with nonpositive flag curvature. We also characterize the Minkowski hyperplanes, Minkowski hyperspheres and Minkowski cylinders as the only hypersurfaces in Minkowski space with parallel second fundamental form.

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.

On stable compact minimal submanifolds

Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the product of two spheres is obtained. Also, it is proved that the only stable compact minimal surfaces of the product of a 2-sphere and any Riemann surface are the complex ones.

On stable compact minimal submanifolds of Riemannian product manifolds

In this paper, we prove a classification theorem for stable compact minimal submanifolds of a Riemannian product of an m1-dimensional (m1≥3) hypersurface M1 in Euclidean space and any Riemannian manifold M2, when the sectional curvature KM1 of M1 satisfies 1m1−1≤KM1≤1. In particular, when the ambient space is an m-dimensional (m≥3) compact hypersurface M in Euclidean space, if the sectional curvature KM of M satisfies 1m+1≤KM≤1, then we conclude that there exist no stable compact minimal submanifolds in M.

Some pinching theorems for minimal submanifolds in $$ \mathbb{S}^m (1) \times \mathbb{R} $$

Let Mn be an n-dimensional compact minimal submanifolds in \( \mathbb{S}^m (1) \times \mathbb{R} \). We prove two pinching theorems by the Ricci curvature and the sectional curvature pinching conditions respectively. In fact, we characterize the Clifford tori and Veronese submanifolds by our pinching conditions respectively.

On Yau rigidity theorem for minimal submanifolds in spheres

In this note, we investigate the well-known Yau rigidity theorem for minimal submanifolds in spheres. Using the parameter method of Yau and the DDVV inequality verified by Lu, Ge and Tang, we prove that if $M$ is an $n$-dimensional oriented compact minimal submanifold in the unit sphere $S^{n+p}(1)$, and if $K_{M}\geq\frac{sgn(p-1)p}{2(p+1)},$ then $M$ is either a totally geodesic sphere, the standard immersion of the product of two spheres, or the Veronese surface in $S^4(1)$. Here $sgn(\cdot)$ is the standard sign function. We also extend the rigidity theorem above to the case where $M$ is a compact submanifold with parallel mean curvature in a space form.

A class of minimal Lagrangian submanifolds in complex hyperquadrics

In this paper we construct a class of compact minimal Lagrangian submanifolds in complex hyperquadrics by studying Gauss maps of compact rotational hypersurfaces in the unit sphere.

The scalar curvature of CR submanifolds of maximal CR dimension of complex projective space

We treat n-dimensional compact minimal submanifolds of complex projective space when the maximal holomorphic tangent subspace is (n − 1)-dimensional and we give a sufficient condition for such submanifolds to be tubes over totally geodesic complex subspaces.

PINCHING THEOREMS FOR A COMPACT MINIMAL SUBMANIFOLD IN A COMPLEX PROJECTIVE SPACE

AbstractWe give a formula for the Laplacian of the second fundamental form of an n-dimensional compact minimal submanifold M in a complex projective space CPm. As an application of this formula, we prove that M is a geodesic minimal hypersphere in CPm if the sectional curvature satisfies K≥1/n, if the normal connection is flat, and if M satisfies an additional condition which is automatically satisfied when M is a CR submanifold. We also prove that M is the complex projective space CPn/2 if K≥3/n, and if the normal connection of M is semi-flat.

The Calabi–Yau conjectures for embedded surfaces

In this talk I will discuss the proof of the Calabi-Yau conjectures for embedded surfaces. This is joint work with Bill Minicozzi, [CM9]. The Calabi-Yau conjectures about surfaces date back to the 1960s. Much work has been done on them over the past four decades. In particular, examples of Jorge-Xavier from 1980 and Nadirashvili from 1996 showed that the immersed versions were false; we will show here that for embedded surfaces, i.e., injective immersions, they are in fact true. Their original form was given in 1965 in [Ca] where E. Calabi made the following two conjectures about minimal surfaces (see also S.S. Chern, page 212 of [Ch]): Conjecture 1. “Prove that a complete minimal hypersurface in R must be unbounded.” Calabi continued: “It is known that there are no compact minimal submanifolds of R (or of any simply connected complete Riemannian manifold with sectional curvature ≤ 0). A more ambitious conjecture is”: Conjecture 2. “A complete minimal hypersurface in R has an unbounded projection in every (n− 2)–dimensional flat subspace.”

Scalar curvature of a class of minimal submanifolds in rank 1 symmetric spaces

In this paper, we give some rigidity theorems which concern with compact minimal coisotropic submanifolds in ℂPn, compact minimal quaternionic coisotropic submanifolds in ℚPn and compact minimal hypersurfaces in P2 (Cay).

A Möbius characterization of submanifolds

In this paper, we study Mobius characterizations of submanifolds without umbilical points in a unit sphere S n + p ( 1 ) . First of all, we proved that, for an n -dimensional ( n ≥ 2 ) submanifold x : M ↦ S n + p ( 1 ) without umbilical points and with vanishing Mobius form Φ , if ( n - 2 ) | | A ˜ | | ≤ n - 1 n n R - 1 n [ ( n - 1 ) 2 - 1 p - 1 ] is satisfied, then, x is Mobius equivalent to an open part of either the Riemannian product S n - 1 ( r ) × S 1 1 - r 2 in S n + 1 ( 1 ) , or the image of the conformal diffeomorphism σ of the standard cylinder S n - 1 ( 1 ) × R in R n + 1 , or the image of the conformal diffeomorphism τ of the Riemannian product S n - 1 ( r ) × H 1 1 + r 2 in H n + 1 , or x is locally Mobius equivalent to the Veronese surface in S 4 ( 1 ) . When p = 1 , our pinching condition is the same as in Main Theorem of Hu and Li [6], in which they assumed that M is compact and the Mobius scalar curvature n ( n - 1 ) R is constant. Secondly, we consider the Mobius sectional curvature of the immersion x . We obtained that, for an n -dimensional compact submanifold x : M ↦ S n + p ( 1 ) without umbilical points and with vanishing form Φ , if the Mobius scalar curvature n ( n - 1 ) R of the immersion x is constant and the Mobius sectional curvature K of the immersion x satisfies K ≥ 0 when p = 1 and K > 0 when p > 1 . Then, x is Mobius equivalent to either the Riemannian product S k ( r ) × S n - k 1 - r 2 , for k = 1 , 2 , ⋯ , n - 1 , in S n + 1 ( 1 ) ; or x is Mobius equivalent to a compact minimal submanifold with constant scalar curvature in S n + p ( 1 ) .

Inequalities for eigenvalues of Laplacian on domains and compact complex hypersurfaces in complex projective spaces

It is well known that the spectrum of Laplacian on a compact Riemannian manifold M is an important analytic invariant and has important geometric meanings. There are many mathematicians to investigate properties of the spectrum of Laplacian and to estimate the spectrum in term of the other geometric quantities of M . When M is a bounded domain in Euclidean spaces, a compact homogeneous Riemannian manifold, a bounded domain in the standard unit sphere or a compact minimal submanifold in the standard unit sphere, the estimates of the k + 1 -th eigenvalue were given by the first k eigenvalues (see [9], [12], [19], [20], [22], [23], [24] and [25]). In this paper, we shall consider the eigenvalue problem of the Laplacian on compact Riemannian manifolds. First of all, we shall give a general inequality of eigenvalues. As its applications, we study the eigenvalue problem of the Laplacian on a bounded domain in the standard complex projective space C P n ( 4 ) and on a compact complex hypersurface without boundary in C P n ( 4 ) . We shall give an explicit estimate of the k + 1 -th eigenvalue of Laplacian on such objects by its first k eigenvalues.