Geometry, PDE and Mathematical Physics
Department of Mathematics and Statistics
Texas Tech University
Today's seminar is also Miraj Samarakkody's Doctoral Dissertation Defense and begins at 2 PM (CST UT-6).
Please watch online via this Zoom link.
Meeting ID: 971 7809 5760
Passcode: Miraj
I will report on joint work with Samuel Pérez-Ayala. We derive a sharp upper bound for the first eigenvalue $\lambda_{1,p}$ of the $p$-Laplacian on asymptotically hyperbolic manifolds for $p$ larger than $1$. We then prove that a particular class of conformally compact submanifolds within asymptotically hyperbolic manifolds are themselves asymptotically hyperbolic. As a corollary, we show that for any minimal conformally compact submanifold $Y^{k+1}$ within $\mathbb{H}^{n+1}(-1)$, $\lambda_{1,p}(Y)=\left(\frac{k}{p}\right)^{p}$. We then obtain lower bounds on the first eigenvalue of these submanifolds in the case where minimality is replaced with a weaker mean curvature assumption and where the ambient space is a general Poincaré-Einstein space whose boundary is of non-negative Yamabe type. In the process, we introduce an invariant $\hat{\beta^Y}$ for each such submanifold, enabling us to generalize a result due to Cheung-Leung.
The emergence of nonlocal theories as promising models in different areas of science (continuum mechanics, biology, image processing) has led the mathematical community to conduct varied investigations of systems of integro-differential equations. In this talk I will present results for systems of integral equations with weakly singular kernels, flux-type boundary conditions, as well as some recent results on nonlocal Helmholtz-Hodge type decompositions with applications at both theoretical and applied levels.
Today's seminar begins at 1 PM (CST UT-6).
Please watch online via this Zoom link.
The Helfrich functional is a type of elastic energy which extends the well-known Willmore energy. The Euler-Lagrange equation describing equilibria is a fourth order nonlinear elliptic partial differential equation the understanding of which is essential to analyze the morphology of cellular membranes.
In this talk, we present a second order reduction of this Euler-Lagrange equation, known as the reduced membrane equation, and exploit its properties to analyze the original fourth order equation.
Starting from an introductory perspective with a presentation of the background behind the problem, we use explicit parametrizations of discrete constant negative Gaussian curvature surfaces (negative CGC) of revolution, i.e. discrete pseudospherical surfaces of revolution, for creating associated families and Bäcklund transformations. This provides new explicit parametrizations of non-rotational discrete pseudospherical surfaces.
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Dynamical systems on 3-manifolds have been paid much attention along the time. In particular, magnetic trajectories are solutions of a second order differential equation (known as the Lorentz equation) and they generalise geodesics. A magnetic field on a Riemannian manifold is defined by a closed 2-form that helps, together with the metric, to define the Lorentz force. On the other hand, magnetic curves derive from the variational problem of the Landau-Hall functional, which is, in the absence of a magnetic field, nothing but the kinetic energy functional.
The dimension 3 is rather special, since it allows us to identify 2-forms with vector fields via the Hodge ⋆ operator and the volume form of the (oriented) manifold. Moreover, in dimension 3, one may define a cross product and therefore, the Lorentz equation may be written in an easier way.
The challenge is to solve the differential equation in order to find an explicit solution, meaning the explicit parametrization for the magnetic trajectories. Nevertheless, this is not always possible and, because of that, it is necessary to understand the behaviour of the solution.
Recent studies are done in 3-dimensional Sasakian manifolds, where the contact 2-form naturally defines a magnetic field. The solutions of the Lorentz equation, usually called contact magnetic trajectories, are often expressed in a concrete parametrization. It can be proved a reduction result for the codimension in a Sasakian space form, that is, essentially, we can reduce the study (of a magnetic curve in a Sasakian space form) to dimension 3. The geometry of magnetic trajectories have been recently studied in the z-sphere, in the Berger 3-sphere, in the Heisenberg group Nil3 and in SL(2,R), respectively.
Another important problem is the existence of closed curves which is a fascinating topic in dynamical systems. Periodic orbits of the contact magnetic fields on the unit 3-sphere and in the Berger 3-sphere were found in the last two decades and conditions for the periodicity have been obtained. A similar result has been recently given; it was proved that periodic contact magnetic curves in SL(2,R) can be quantized in the set of rational numbers.
The geometry of contact magnetic curves in SL(2,R) is much more beautiful. More precisely, it can be shown that every contact magnetic trajectory (of charge q) starting at the origin of SL(2,R) with initial velocity X and with charge q is the product of the homogeneous geodesic with initial velocity X and the charged Reeb flow exp(2qs\xi). Similar results are obtained for the Berger spheres as well.
This talk is based on several papers, mainly in collaboration with Prof. Jun-ichi Inoguchi (Japan).
Please watch online via this Zoom link.
Abstract PDF
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The shape of the Red Blood Cell (RBC) exhibits a unique toric-like shape and has posed challenges in mathematical modelling. In the 1970’s, Peter Canham and Wolfgang Helfrich pioneered techniques from elastic energy theory, reframing the study of cell membranes via the integral of the mean curvature squared over a surface called the Helfrich functional, $$\int\int_M \beta(H-c_o)^2 + K dS +\int \lambda dS + \int \Delta P dV$$ where $c_o$ is the spontaneous curvature, $\Delta P$ the partial pressure, $\lambda $ the surface tension, and $\beta$ the bending rigidity respectively. Varying this functional yields a fourth-order elliptic PDE for equilibrium membrane shapes.
Later, Canham introduced the now-standard “axisymmetric model” which reframes the Euler-Lagrange equation into a nonlinear second order ODE known as the shape equation. This model describes rotationally symmetric surfaces generated from a profile curve, parametrized in terms of a rotational radius $r$, and the angle $\psi(r)$ between the profile’s tangent and a horizontal line.
Cassini ovals, a family of toric-like curves, have been proposed as apt RBC models. We show that these curves fail to satisfy the shape equation, and therefore the Euler-Lagrange equation, ruling them out as energy minimizing solutions to the Helfrich energy functional.
The notion of great antipodal sets of a Riemannian manifold was introduced in [B.-Y. Chen and T. Nagano, Un invariant geometrique riemannien, C.R. Acad. Sci. Paris Math. 295 (1982), 389–391]. Later, great antipodal sets have been studied by a number of mathematicians and was shown that great antipodal sets relate closely with several important areas in mathematics.
In this talk, I will present this geometric notion and explain how this geometric notion can be applied to several important areas in mathematics.
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