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Genuine infinitesimal bendings of Euclidean submanifolds

Miguel Ibieta Jiménez IMPA

In this talk we focus on a notion of bending of a submanifold. This notion is associated to variations of a submanifold by immersions that preserve lengths “up to the first order”. More precisely, an infinitesimal bending of an isometric immersion $f : M^n \to \mathbb{R}^{n+p}$ is the variational vector field associated to a variation of $f = f_0$ by immersions $f_t$ whose induced metrics $g_t$ satisfy $g\prime{}_{t}(0) = 0$.
We give a description of the complete Euclidean hypersurfaces that admit non-trivial infinitesimal bendings. We also present some results concerning genuine infinitesimal bendings of submanifolds in low codimension. That an infinitesimal bending is genuine means that it is not determined by an infinitesimal bending of a submanifold of larger dimension. We show that a strong local condition for a submanifold to be genuinely infinitesimally bendable is to be ruled and we estimate the dimension of the rulings. Finally, we describe the situation for infinitesimal bendings of compact submanifolds in codimension 2.
This is a joint work with M. Dajczer.

Teoremas de semiespacio para superficies con curvatura media predeterminada

Motivados por el teorema de semiespacio clásico de Hoffman y Meeks para superficies mínimas en $\mathbb{R}^3$, el objetivo de esta charla es obtener resultados de tipo semiespacio para superficies inmersas en el espacio Euclideo $\mathbb{R}^3$ cuya curvatura media viene dada por una función predeterminada dependiendo de su aplicación de Gauss.

Seminario 1ª planta, IEMath-GR

Conformal Killing Initial Data

Igor Khavkine Institute of Mathematics of the Czech Academy of Sciences, Praga

We find necessary and sufficient conditions ensuring that the vacuum development of an initial data set of the Einstein's field equations admits a conformal Killing vector. We refer to these conditions as conformal Killing initial data (CKID) and they extend the well-known Killing initial data that have been known for a long time. The procedure used to find the CKID is a classical argument based on the computation of a suitable propagation identity. The propagation identity is valid in any pseudo-Riemannian signature. In Lorentzian signature it involves hyperbolic operators, while in Riemannian it involves elliptic ones, which may be of independent interest. (Based on arXiv:1905.01231, with A. García-Parrado.)

Seminario 1ª planta, IEMath

Despacho: Iemath

William H. Meeks III

University of Massachusetts, Amherst

Despacho: 4, segunda planta

Marcos Paulo Tassi

Despacho: IEMath, B1-4

Despacho: IEMath

Miguel Ibieta Jiménez

IMPA

Despacho: IEMath D9

Despacho: IEMath

Despacho: IEMAth

José M. M. Senovilla

Despacho: IEMath D9

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