Fluxo de curvatura média e hipersuperfícies Tipo-T-Einstein
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2023-05-12
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Universidade Federal de Goiás
Resumo
We present an analysis of the self-similar solutions of Mean Curvature Flow (MCF) by
ruled and revolution surfaces in $\mathbb{R}^{3}$. We prove that homothetic helicoidal
motion solutions whose initial condition is a non-cylindrical ruled surface must be trivial. When
the initial condition is a surface of revolution, we characterize the solutions in terms of the
curvature of the generatrix curve.
We characterize the curve shortening flow (CSF) soliton solutions on the torus of
revolution $\mathbb{T}^{2}\subset\mathbb{R}^3$. We show that the solutions must be
asymptotic to the equators of the torus. Furthermore, we generalize this result to surfaces of
revolution in $\mathbb{R}^3$.
Finally, we prove that a class of Einstein-type hypersurfaces in $\mathbb{S}^n \times
\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$ are rotational or totally umbilical
hypersurfaces.
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Citação
NOVAIS, R. M. Fluxo de curvatura média e hipersuperfícies Tipo-T-Einstein. 2023. 105 f. Tese (Doutorado em Matemática) - Universidade Federal de Goiás, Goiânia, 2023.