Probability, Differential Geometry and Mathematical Physics
Department of Mathematics and Statistics
Texas Tech University
We explore the Enneper family of real maximal surfaces using Weierstrass data in three-dimensional Minkowski space. Our focus is on deriving the algebraic surfaces within this family and analyzing their class and degree.
This seminar may be viewed in the TTU Mediasite Catalog and the slideshow pdf is available.
The existence of embedded geodesics on surfaces is the foundational problem. I will explain the existence of two capillary embedded geodesics on Riemannian 2-disks with a strictly convex boundary with a certain condition via multi-parameter min-max construction. I will then explain the existence of two free boundary embedded geodesics on Riemannian 2-disks with a strictly convex boundary by free boundary curve shortening flow on surfaces, which is a free boundary analog of Grayson’s theorem in 1989. Finally, I will explain the Morse Index bound of such geodesics.
Watch online via this Zoom link.
We present the fourth fundamental form and curvature formulas for hypersurfaces in four-dimensional Euclidean space. These quantities are defined and computed for rotational hypersurfaces. Furthermore, we investigate rotational hypersurfaces that satisfy a specific relation involving the fourth Laplacian and a 4X4 matrix.
This seminar may be viewed in the TTU Mediasite Catalog and the slideshow pdf is available.
It is well known that four-dimensional Riemannian manifolds carry many peculiar properties, which give rise to the existence of unique canonical metrics (e.g. half conformally flat metrics). In their study of self-dual solutions of Yang-Mills equations, Atiyah, Hitchin and Singer adapted the celebrated Penrose’s construction of twistor spaces to the Riemannian context, showing that a Riemannian four-manifold is half conformally flat if and only if its twistor space is a complex manifold: this paved the way for the study of many other characterizations of curvature properties for Riemannian four-manifolds. After giving an overview of the Riemannian and Hermitian structures of twistor spaces in the four-dimensional case, we present some new rigidity results for Riemannian four-manifolds whose twistor spaces satisfy specific vanishing curvature conditions. We also address the problem of classifying Einstein four-manifolds with positive sectional curvature, establishing a partial result obtained via twistor methods. This is based on joint work with Giovanni Catino and Paolo Mastrolia.
Watch online Tuesday at 4 PM CST (UT-6) via this Zoom link.
For a closed embedded submanifold $\Sigma^n$ of a closed Riemannian manifold $(M^{n+k}, g)$, with $k < n + 2$, we define extrinsic global conformal invariants of $\Sigma$ by renormalizing the volume associated to the unique singular Yamabe metric with singular set $\Sigma$. For odd $n$, the renormalized volume is an absolute conformal invariant, while for even $n$, there is a conformally invariant energy term given by the integral of a local Riemannian invariant. We also compute the derivatives of these quantities with respect to variations of the submanifold. We compare these results with their counterparts in the CCE and the classical singular Yamabe contexts.
For even-dimensional submanifolds, the notion of energy extends to most codimensions without the dimensionality constraint when viewed from a formal standpoint, allowing for an introduction of a larger class of conformal invariants. Even in the exceptional codimensions, interesting behaviors arise.
This talk is based on my doctoral work under the supervision of Stephen McKeown.
Available online via this Zoom link.