Julien Roth

Spinorial characterizations of surfaces into three-dimensional homogeneous manifolds

Abstract We give spinorial characterizations of isometrically immersed surfaces into three-dimensional homogeneous manifolds with four-dimensional isometry group in terms of the existence of a particular spinor field. This generalizes works by Friedrich for R 3 and Morel for S 3 and H 3 . The main argument is the interpretation of the energy–momentum tensor of such a spinor field as the second fundamental form up to a tensor depending on the structure of the ambient space.

Spinorial Characterizations of Surfaces into 3-dimensional Pseudo-Riemannian Space Forms

We give a spinorial characterization of isometrically immersed surfaces of arbitrary signature into 3-dimensional pseudo-Riemannian space forms. This generalizes a recent work of the first author for spacelike immersed Lorentzian surfaces in ℝ2,1 to other Lorentzian space forms. We also characterize immersions of Riemannian surfaces in these spaces. From this we can deduce analogous results for timelike immersions of Lorentzian surfaces in space forms of corresponding signature, as well as for spacelike and timelike immersions of surfaces of signature (0, 2), hence achieving a complete spinorial description for this class of pseudo-Riemannian immersions.

ISOMETRIC IMMERSIONS INTO LORENTZIAN PRODUCTS

We give a necessary and sufficient condition for an n-dimensional Riemannian manifold to be isometrically immersed into one of the Lorentzian products 𝕊n × ℝ1 or ℍn × ℝ1. This condition is expressed in terms of its first and second fundamental forms, the tangent and normal projections of the vertical vector field. As applications, we give an equivalent condition in a spinorial way and we deduce the existence of a one-parameter family of isometric maximal deformation of a given maximal surface obtained by rotating the shape operator.

Pinching of the first eigenvalue for second order operators on hypersurfaces of the Euclidean space

We prove stability results associated with upper bounds for the first eigenvalue of certain second order differential operators of divergence-type on hypersurfaces of the Euclidean space. We deduce some applications to r-stability as well as to almost-Einstein hypersurfaces.

A remark on almost umbilical hypersurfaces

In this article, we prove new stability results for almost-Einstein hypersurfaces of the Euclidean space, based on previous eigenvalue pinching results. Then, we deduce some comparable results for almost umbilical hypersurfaces.

Spinorial representation of submanifolds in Riemannian space forms

In this paper we give a spinorial representation of submanifolds of any dimension and codimension into Riemannian space forms in terms of the existence of so called generalized Killing spinors. We then discuss several applications, among them a new and concise proof of the fundamental theorem of submanifold theory. We also recover results of T. Friedrich, B. Morel and the authors in dimension 2 and 3.

A FUNDAMENTAL THEOREM FOR SUBMANIFOLDS OF MULTIPRODUCTS OF REAL SPACE FORMS

Abstract We prove a Bonnet theorem for isometric immersions of submanifolds into the products of an arbitrary number of simply connected real space forms. Then we prove the existence of associate families of minimal surfaces in such products. Finally, in the case of 𝕊2 × 𝕊2, we give a complex version of the main theorem in terms of the two canonical complex structures of 𝕊2 × 𝕊2.

A NEW RESULT ABOUT ALMOST UMBILICAL HYPERSURFACES OF REAL SPACE FORMS

In this short note, we prove that an almost umbilical hypersurface of a real space form with almost Codazzi umbilicity tensor is embedded, diffeomorphic and quasi-isometric to a round sphere. Then, we derive a new characterisation of geodesic spheres in space forms. DOI: 10.1017/S0004972714000732

Spinorial representation of submanifolds in metric Lie groups

Abstract In this paper we give a spinorial representation of submanifolds of any dimension and codimension into Lie groups equipped with left invariant metrics. As applications, we get a spinorial proof of the Fundamental Theorem for submanifolds into Lie groups, we recover previously known representations of submanifolds in R n and in the 3-dimensional Lie groups S 3 and E ( κ , τ ) , and we get a new spinorial representation for surfaces in the 3-dimensional semi-direct products: this achieves the spinorial representations of surfaces in the 3-dimensional homogeneous spaces. We finally indicate how to recover a Weierstrass-type representation for CMC-surfaces in 3-dimensional metric Lie groups recently given by Meeks, Mira, Perez and Ros.

A DDVV inequality for submanifolds of warped products

We prove a DDVV inequality for submanifolds of warped products of the form I ×a M n (c) where I is an interval and M n (c) a real space form of curvature c. As an application, we give a rigidity result for submanifolds of R × e λt H n (c).