Differential Geometry, PDE and Mathematical Physics
Department of Mathematics and Statistics
Texas Tech University
The Differential Geometry, Partial Differential Equations and Mathematical Physics seminar group recommends people attend the Colloquium given on the 9th in Experimental Sciences Building 1, room 120.
A class of space-like rotational hypersurfaces, represented by the parametrization $\mathbf{x}(u, v, w)$, is investigated within the four-dimensional pseudo-Euclidean space $\mathbb{E}^4_2$. The curvatures of the hypersurface are derived. In addition, the associated Laplace--Beltrami operator is computed, and it is shown that the hypersurface satisfies the eigenvalue equation $\Delta \mathbf{x} = \mathcal{A} \mathbf{x}$, where $\mathcal{A}$ is a $4 \times 4$ matrix.
In 1983, Costa discovered a complete embedded minimal surface in three-dimensional Euclidean space with genus one and three embedded ends. Building on this result, Hoffman and Meeks later constructed complete embedded minimal surfaces with three embedded ends and arbitrary genus. These surfaces can be viewed as desingularizations of the union of a catenoid and a plane along their intersection circle. In this talk, we aim to explore analogous constructions in four-dimensional Euclidean space. Specifically, we discuss the desingularization of the union of a Lagrangian catenoid and a two-dimensional plane, highlighting both the similarities to and the differences from the Costa–Hoffman–Meeks surfaces in three-dimensional space.
This Differential Geometry, PDE and Mathematical Physics seminar is available over zoom.
We study a class of rotational hypersurfaces with five parameters in the six-dimensional Euclidean space $\mathbb{E}^6$. We derive the associated curvature functions and examine the geometric properties of these hypersurfaces. Furthermore, we apply the Laplace--Beltrami operator and determine the conditions under which the relation $\Delta \mathbf{x} = \mathcal{B} \mathbf{x}$ holds, where $\mathcal{B}$ is a $6 \times 6$ matrix.
In recent decades, a series of eigenvalue estimates have been established for minimal-type hypersurfaces under geometric constrains. A foundational result was given by Choi and Wang (1983), who proved a lower bound for the first eigenvalue of closed minimal surfaces in complete Riemannian manifolds with positive Ricci curvature. Their wok was later extended by Cheng-Mejia-Zhou and Ding-Xin to closed f-minimal surfaces and closed self-shrinkers, respectively.
Beyond the closed case, Brendle and Tsiamis considered complete non-compact self-shrinkers, proving that the first eigenvalue admits a universal lower bound 1/4. More recently, Conrado and Zhou generalized this direction to f-minimal hypersurfaces in gradient shrinking Ricci solitons, establishing not only lower bounds but also the discreteness of the spectrum of the drifted Laplacian.
In this talk, I will present a further generalization to λ-hypersurfaces in R^{n+1}. Under the condition |λ|≤1/2-1/(2n), we show that the first eigenvalue is bounded below by 1/4- λ^2/2. This estimate recovers the result of Brendle-Tsiamis in the special case λ=0. Since self-shrinkers and λ-hypersurfaces can be regarded as minimal and constant mean curvature(CMC) hypersurfaces in a smooth metric measure space with weight function |x|^2/4, our result provides a natural extension of eigenvalue estimates from the minimal to the CMC setting.
This Differential Geometry, PDE and Mathematical Physics seminar is available over zoom.
 | Tuesday
Nov. 4
4 PM
MA 010
|
| TBA
Cihan Özgür
Mathematics, İzmir Demokrasi Üniversitesi
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To be announced
To be announced