| Time | Speaker | Title | Resources | |
|---|---|---|---|---|
| 10:30 to 11:40 | Rahul Kumar Singh (Indian Institute of Technology Patna, India) |
Zero Mean Curvature (ZMC) surfaces and Euler-Ramanujan's identities We discuss a connection between some special zero mean curvature surfaces and certain Euler-Ramanujan's identities. |
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| 15:00 to 16:10 | Edmund Adam Paxton (University of Oxford, United Kingdom) |
Embeddedness of timelike maximal surfaces in (1+2)-Minkowski Space Timelike maximal surfaces (i.e. zero mean curvature surfaces) in Minkowski space R^{1+2} are the hyperbolic analogue of minimal surfaces in Euclidean space R^3. They have been suggested as models for some physical theories and provide interesting problems in geometric analysis and wave equations.
To set the stage, I will begin the talk with a (very) brief survey of some famous results for minimal surfaces in R^3, before giving a gentle introduction to the picture in R^{1+2}. I will place particular emphasis here on the Cauchy problem for timelike maximal surfaces in R^{1+2}---which may be regarded as the “natural” setting, given that the underlying equations are hyperbolic---and I will present some examples and well-known results to illustrate this theory. In the main part of the talk I will then present some recent results on the global geometry of timelike maximal surfaces in R^{1+2} and on singularity formation for the Cauchy evolution. To be precise, I will present two main results:
Theorem 1. Every smooth properly immersed timelike maximal surface in R^{1+2} is embedded, and is a smooth graph over bounded subsets.
Theorem 2. Singularity formation for a timelike maximal surface always involves a curvature blow up, with the blow up occurring in an L^1-L^\infty norm.
and I will sketch a proof of Theorem 1. I will also very briefly touch upon the existence of (global) $C^1$ solutions to the Cauchy evolution in the case that $C^2$ solutions fail (dues to Theorems 1 & 2).
The talk will follow closely the content of the paper https://arxiv.org/abs/1902.08952
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