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Minimal surfaces in the round 3-sphere enjoy a number of attractive and influential rigidity properties, particularly in low genus. Progress has been made in extending these results to certain analogous settings, such as free boundary minimal surfaces in the Euclidean ball. We will propose perspectives by which these other settings may be studied within a continuous family starting with the round 3-sphere. In this talk, we’ll discuss (half-space) intersection properties, connections with capillary minimal surfaces and other free boundary problems. This is based on joint work (including some in progress) with Keaton Naff.

Complete minimal surfaces in R^3 have been much studied but much less is known about R^4. I will recall the main tools in R^4 and give a couple of examples of minimal embeddings of the complex plane in R^4. Then I will focus on complete minimal tori of curvature -8π and one end: in R^3 there is a unique example (the Chen-Gackstetter square torus) but in R^4 we can construct examples on all the rectangular tori. I will discuss the strategy for the construction and indicate the many questions which remain open. Joint work with Marc Soret.

We discuss the topological realization problem for minimally embedding compact surfaces in round spheres and balls. In 1970, using the solution to the Plateau problem, Lawson constructed orientable minimal surfaces of each genus embedded in $S^3$. In recent work with Karpukhin, McGrath and Stern, using equivariant eigenvalue optimization methods and a priori eigenspace dimension bounds, we constructed orientable free boundary minimal surfaces in $B^3$ of any genus and (positive) number of boundary, components. Now we have extended our methods to handle the nonorientable case, constructing embedded minimal surfaces in $S^4$ diffeomorphic to the connect-sum of any number of real projective planes: these all have area (so Willmore bending energy) under $8\pi$ and enjoy other interesting geometric features. The analogous construction for nonorientable free boundary minimal surfaces embedded in $B^4$ looms on the horizon. Open problems and speculation about potential discretizations may be offered if time permits....

We will talk of interpolation problems of two types.

First type of interpolation we talk of is that given two real analytic curves can one interpolate them with a minimal or maximal surface or a CMC surface? -- a version of Plateau's problem. For minimal surfaces this problem was solved by Douglas and Rado in great generality. We show that indeed, if the curves are "close" enough in a certain sense, then interpolation is possible. We will also talk of existence of a maximal surface containing a given real analytic curve and a special singularity, under certain conditions.

The second type of interpolation we will talk about is given a array of surfaces placed at some periodic intervals, can one interpolate them by a minimal/maximal surface, in the sense that the height functions of surfaces at these arrays sum up to a height function of a minimal/maximal surface.This work uses some Euler-Ramanujan identities.

Discrete minimal surfaces in the Euclidean space are central in the research field discrete differential geometry. Similarly, we can consider discrete spacelike maximal surfaces and discrete timelike minimal surfaces in Lorentz-Minkowski space. Although their formulation is analogous to discrete minimal surfaces, their behaviors are quite different.

In this talk we introduce recent progress on discrete zero mean curvature surfaces in Euclidean and Lorentz-Minkowski spaces. After briefly introducing basic results on discrete minimal surfaces, we investigate discrete zero mean curvature surfaces in Lorentz-Minkowski space and their singular behaviors. Furthermore, if time permits, we will introduce a construction of discrete zero mean curvature surfaces in Lorentz-Minkowski space that change causal characters along specific singularities.

This talk is partially based on ongoing project with Joseph Cho, Wayne Rossman, and Seong-Deog Yang.

This talk presents a study of biconservative hypersurfaces M4r(r = 0, 1, 2, 3, 4) in the pseudo-Euclidean space E5s(s = 0, 1, 2, 3, 4, 5) with constant norm of the second fundamental form. We find that every such hypersurface in E5s with a diagonal shape operator has constant mean curvature and constant scalar curvature. This is a joint work with Ram Shankar Gupta, Andreas Arvanitoyeorgos, and Marina Statha.

There is a broad body of work devoted to proving theorems of the following form: spaces with infinitely many special sub-spaces are either nonexistent or rare. Such finiteness statements are important in algebraic geometry, number theory, and the theory of moduli space and locally symmetric spaces. I will talk about joint work with Simion Filip and David Fisher proving a finiteness statement of this kind in a differential geometry setting. Our main theorem is that a closed negatively curved analytic Riemannian manifold with infinitely many closed totally geodesic hypersurfaces must be isometric to an arithmetic hyperbolic manifold. The talk will be more focused on providing background and context than details of proofs and should be accessible to a general audience.

CMC (constant mean curvature) surfaces are critical with respect to area as long as we restrict to competitor surfaces which 'enclose the same volume'. The index of a CMC surface is the number of ways we can locally deform it to reduce its area whilst enclosing the same amount of volume. We will show that the index of a CMC surface is bounded linearly from above in terms of its genus and a Willmore-type energy i.e. the more unstable a surface is the more genus and/or ‘Willmore energy’ it must have. Joint work with Luca Seemungal. 

For a closed Riemann surface X, and an irreducible representation R from the fundamental group of X to PSL(2,C), a seminal theorem of Donaldson proves the existence of an R-equivariant harmonic map from the universal cover of X into hyperbolic 3-space. We shall discuss a generalization of this result to the case when X is a punctured Riemann surface arising from an element of the enhanced Teichmüller space, and R is a framed PSL(2,C)-representation. Our technique involves the harmonic map heat flow, and the relation between the asymptotic behaviour of a harmonic map and the singular-flat geometry of its Hopf differential. This is joint work with Gobinda Sau.