Cornell Analysis Seminar

Title:  Existence of constant mean curvature surfaces with bounded genus in general 3-manifolds. 

Abstract: In this talk, I will introduce some history on finding minimal surfaces and constant mean curvature (CMC) surfaces. Then, I will talk about our recent work on the construction of CMC surfaces with genus control in a general 3-manifold. Our construction is based on a penalized area functional introduced by Riviere. This talk is based on joint work with Filippo Gaia. 

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Title: Min-max barrier and minimal foliations on the torus. 

Abstract: Aubry–Mather theory concerns minimizing orbits of convex Hamiltonian systems. Moser, Bangert, and Auer developed a higher-dimensional analogue for area-minimizing hypersurfaces. In this talk, I will describe this theory on a Riemannian torus. After lifting to the universal cover, each minimizing hypersurface carries an asymptotic invariant, called its homological direction, and hypersurfaces with a common direction organize into laminations. I will discuss recent work in which Almgren–Pitts min–max theory is used to characterize when such a lamination is a foliation. I will also explain how, for generic metrics, a gap in the lamination contains a complete embedded minimal hypersurface that is not area-minimizing. 

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