Special Structures in Riemannian Geometry

There are two natural classes of Einstein metrics on bounded domains. (I) metrics which extend smoothly to the boundary, and (II) complete metrics which conformally extend to the boundary, (conformally compact metrics). We will discuss similarities and differences on the structure of these spaces of Einstein metrics, in particular in regard to the “natural” boundary value problems. Speaker: Adrian Butscher (Stanford University) Title: “Gluing Constructions for Constant Mean Curvature Surfaces” Abstract: I will review the now classical Kapouleas gluing construction for CMC surfaces in Euclidean space and present some results and work in progress concerning the extensions of this theory to general ambient manifolds. An important feature which emerges is that the ambient Riemannian curvature seems to play a significant role in the existence of such surfaces; and exploiting this, it seems possible to construct examples of CMC surfaces having properties very different from their Euclidean analogues. I will review the now classical Kapouleas gluing construction for CMC surfaces in Euclidean space and present some results and work in progress concerning the extensions of this theory to general ambient manifolds. An important feature which emerges is that the ambient Riemannian curvature seems to play a significant role in the existence of such surfaces; and exploiting this, it seems possible to construct examples of CMC surfaces having properties very different from their Euclidean analogues. Speaker: Benoit Charbonneau (Duke University) Title: “Existence of periodic instantons” Abstract: Yang–Mills instantons on S1 × R3 (often called calorons) are in correspondence, via the Nahm transform, to solutions to Nahm’s equations on the circle. In joint work with Jacques Hurtubise, we completed Nye and Singer’s proof of this Nahm transform correspondence. We also proved that the solutions on the circle are in correspondence, by a twistor transform, to certain classes of vector bundles on an associated twistor space. Those correspondence allow us to compute the moduli space of these objects, settling some very natural existence questions. Yang–Mills instantons on S1 × R3 (often called calorons) are in correspondence, via the Nahm transform, to solutions to Nahm’s equations on the circle. In joint work with Jacques Hurtubise, we completed Nye and Singer’s proof of this Nahm transform correspondence. We also proved that the solutions on the circle are in correspondence, by a twistor transform, to certain classes of vector bundles on an associated twistor space. Those correspondence allow us to compute the moduli space of these objects, settling some very natural existence questions.