Spacelike translating solitons of the mean curvature flow in Lorentzian product spaces with density

Márcio Batista; Giovanni Molica Bisci; Henrique de Lima; Márcio Batista; Giovanni Molica Bisci; Henrique de Lima

doi:10.3934/mine.2023054

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-18. doi: 10.3934/mine.2023054

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Spacelike translating solitons of the mean curvature flow in Lorentzian product spaces with density

1.
CPMAT-IM, Universidade Federal de Alagoas, 57072-970 Maceió, Alagoas, Brazil
2.
Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino Carlo Bo, Piazza della Repubblica 13, 61029 Urbino (Pesaro e Urbino), Italy
3.
Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paraíba, Brazil

Received: 12 May 2022 Revised: 24 July 2022 Accepted: 26 September 2022 Published: 12 October 2022

By applying suitable Liouville-type results, an appropriate parabolicity criterion, and a version of the Omori-Yau's maximum principle for the drift Laplacian, we infer the uniqueness and nonexistence of complete spacelike translating solitons of the mean curvature flow in a Lorentzian product space $ \mathbb R_1\times\mathbb P^n_f $ endowed with a weight function $ f $ and whose Riemannian base $ \mathbb P^n $ is supposed to be complete and with nonnegative Bakry-Émery-Ricci tensor. When the ambient space is either $ \mathbb R_1\times\mathbb G^n $, where $ \mathbb G^n $ stands for the so-called $ n $-dimensional Gaussian space (which is the Euclidean space $ \mathbb R^n $ endowed with the Gaussian probability measure) or $ \mathbb R_1\times\mathbb H_f^n $, where $ \mathbb H^n $ denotes the standard $ n $-dimensional hyperbolic space and $ f $ is the square of the distance function to a fixed point of $ \mathbb H^n $, we derive some interesting consequences of our uniqueness and nonexistence results. In particular, we obtain nonexistence results concerning entire spacelike translating graphs constructed over $ \mathbb P^n $.
- weighted Lorentzian product spaces,
- Bakry-Émery-Ricci tensor,
- drift Laplacian,
- $ f $-mean curvature,
- mean curvature flow,
- spacelike translating solitons,
- Gaussian space
Citation: Márcio Batista, Giovanni Molica Bisci, Henrique de Lima. Spacelike translating solitons of the mean curvature flow in Lorentzian product spaces with density[J]. Mathematics in Engineering, 2023, 5(3): 1-18. doi: 10.3934/mine.2023054

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Abstract

By applying suitable Liouville-type results, an appropriate parabolicity criterion, and a version of the Omori-Yau's maximum principle for the drift Laplacian, we infer the uniqueness and nonexistence of complete spacelike translating solitons of the mean curvature flow in a Lorentzian product space $ \mathbb R_1\times\mathbb P^n_f $ endowed with a weight function $ f $ and whose Riemannian base $ \mathbb P^n $ is supposed to be complete and with nonnegative Bakry-Émery-Ricci tensor. When the ambient space is either $ \mathbb R_1\times\mathbb G^n $, where $ \mathbb G^n $ stands for the so-called $ n $-dimensional Gaussian space (which is the Euclidean space $ \mathbb R^n $ endowed with the Gaussian probability measure) or $ \mathbb R_1\times\mathbb H_f^n $, where $ \mathbb H^n $ denotes the standard $ n $-dimensional hyperbolic space and $ f $ is the square of the distance function to a fixed point of $ \mathbb H^n $, we derive some interesting consequences of our uniqueness and nonexistence results. In particular, we obtain nonexistence results concerning entire spacelike translating graphs constructed over $ \mathbb P^n $.

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