Let M be an n ( > 2 ) -dimensional closed orientable submanifold in an ( n + p ) -dimensional space form R n + p ( c ) . We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on M defined by L T f = − div ( T ∇ f ) , where T is a general symmetric, positive definite and divergence-free ( 1 , 1 ) -tensor on M. The upper bound is given in terms of an integration involving tr T and | H T | 2 , where tr T is the trace of the tensor T and H T = ∑ i = 1 n A ( T e i , e i ) is a normal vector field associated with T and the second fundamental form A of M. Furthermore, we give the sufficient and necessary conditions when the upper bound is attained. Our main theorem can be viewed as an extension of the famous “Reilly inequality”. The operator L T can be regarded as a natural generalization of the well-known operator L r which is the linearized operator of the first variation of the ( r + 1 ) -th mean curvature for hypersurfaces in a space form. As applications of our main theorem, we generalize the results of Grosjean [17] and Li–Wang [20] in codimension one to arbitrary codimension.

Sharp Reilly-type inequalities for a class of elliptic operators on submanifolds

Submanifolds with constant scalar curvature in a space form

We extend a sharp upper bound of the first strong stability eigenvalue recently due to Chen and Cheng 
(2016) for the context of closed submanifold $$M^n$$
 immersed with nonzero parallel mean curvature vector field in the Euclidean unit sphere $$\mathbb S^{n+p}$$
 and, as a consequence of this estimate, we obtain new characterizations for certain Clifford tori.

On the First Strong Stability Eigenvalue of Closed Submanifolds in the Unit Sphere

In this paper, we deal with complete linear Weingarten submanifolds M n immersed with parallel normalized mean curvature vector field in a Riemannian space form Q c n + p of constant sectional curvature c and with n ≥ 4 . We recall that a submanifold is called linear Weingarten when its mean and scalar curvatures are linearly related. In this setting, we establish a suitable extension of the generalized maximum principle of Omori–Yau in order to show that such a submanifold M n must be either totally umbilical or isometric to a Clifford torus S 1 ( 1 − r 2 ) × S n − 1 ( r ) , when c = 1 , a circular cylinder R × S n − 1 ( r ) , when c = 0 , or a hyperbolic cylinder H 1 ( − 1 + r 2 ) × S n − 1 ( r ) , when c = − 1 . We also study the parabolicity of these submanifolds with respect to a Cheng–Yau modified operator.

Characterizations of linear Weingarten submanifolds

A Simons type formula for submanifolds with parallel normalized mean curvature vector field (pnmc submanifolds) in the product spaces S n × R , where S n is the unit Euclidean sphere, is proved. As an application, an integral inequality for pnmc closed submanifolds in S n × R is obtained. By this, it is shown that the sharpness in this inequality is attained, in the totally umbilical hypersurfaces, and in a certain family of standard tori of the form S 1 ( 1 − r 2 ) × S n − 1 ( r ) with 0 r 1 .

A Simons type integral inequality for closed submanifolds in the product space Sn×R

We study the submanifolds in the unit sphere ${\boldsymbol S}^{n+p}$ with constant scalar curvature and parallel normalized mean curvature vector field. In this case, we can generalize the work of the second author about hypersurfaces in Hypersurfaces with constant scalar curvature in space forms to submanifolds in a unit sphere.

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Xi Guo

Papers

Submanifolds with constant scalar curvature in a unit sphere