# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### A conformally invariant gap theorem in Yang-Mills theory

We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem. This is joint work with Jeffrey Streets (UC Irvine) and Matthew Gursky (University of Notre Dame).

##### Almost Rigidity of the Positive Mass Theorem

The Positive Mass Theorem states that an asymptotically flat Riemannian manifold, $M^3$, with nonegative Scalar curvature has nonnegative ADM mass, $m_{ADM}(M)\ge 0$, and if the ADM mass is 0 then we have rigidity: the manifold is isometric to Euclidean space. It has long been known that if one has a sequence of such manifolds $M^3_j$ with $m_{ADM}(M_j) \to 0$ then $M_j$ need not converge smoothly to Euclidean space. To avoid bubbling, one forbids the sequence of manifolds to have closed interior minimal surfaces, but allows the manifolds to have minimal boundaries. Dan Lee and I conjectured that in this setting the $M_j$ converge to Euclidean space in the pointed intrinsic flat sense for well chosen points. We proved this in the rotationally symmetric setting and provided examples in that setting where smooth and Gromov-Hausdorff

##### Min-max theory for constant mean curvature hypersurfaces

We describe the construction of closed constant mean curvature (CMC) hypersurfaces using min-max methods. In particular, our theory allows us to show the existence of closed CMC hypersurfaces of any prescribed mean curvature in any closed Riemannian manifold. This work is joint with Xin Zhou.

##### Compactification of the configuration space for constant curvature conical metrics

In this joint work with Rafe Mazzeo, we would like to understand the deformation theory of constant curvature metrics with prescribed conical singularities on a compact Riemann surface. We construct a resolution of the configuration space, and prove a new regularity result that the family of constant curvature conical metrics has a nice compactification as the cone points coalesce. This is one key ingredient to understand the full moduli space of such metrics with positive curvature and cone angles bigger than $2\pi$.

##### Deformations of Q-Curvature

Stability (local surjectivity) and rigidity of the scalar curvature have been studied in an early work of Fischer-Marsden on \vacuum static spaces". Inspired by this line of research, we seek similar properties for Q-curvature by studying \Q-singular spaces", which were introduced by Chang-Gursky-Yang. In this talk, we investigate deformation problems of Q-curvature on closed Rie- mannian manifolds with dimensions n 3. In particular, we classify nonnegative Einstein Q-singular spaces and prove local surjectivity for non-Q-singular spaces. We also prove local rigidity of at manifolds. For global results, we show that any smooth functions can be realized as a Q- curvature on generic Q- at manifolds. However, a locally conformally at metric on n-tori with nonnegative Q-curvature has to be at. This is joint work with Wei Yuan.

##### Willmore Stability of Minimal Surfaces in Spheres

Minimal surfaces in the round n-sphere are prominent examples of surfaces critical for the Willmore bending energy W; those of low area provide candidates for W-minimizers. To understand when such surfaces are W-stable, we study the interplay between the spectra of their Laplace-Beltrami, area-Jacobi and W-Jacobi operators. We use this to prove: 1) the square Clifford torus in the 3-sphere is the only W-minimizer among 2-tori in the n-sphere; 2) the hexagonal Itoh-Montiel-Ros torus in the 5-sphere is the only other W-stable minimal 2-torus in the n-sphere, for all n; 3) the Itoh-Montiel-Ros torus is a local minimum for the conformally-constrained Willmore problem, evidence for a recent conjecture of Lynn Heller and Franz Pedit. We also give sharp estimates on the Morse index of the area for minimal 2-tori in the n-sphere.

##### Singular Volume Problems

We give general results for the asymptotics and anomaly for volumes of regions with respect to measures that are singular along hypersurfaces. These have applications to holography in physical models as well as invariant theory for embedded hypersurfaces. In both cases one solves an appropriate bulk problem with boundary data specified along the singular hypersurface. In particular we focus on applications to the conformal geometry of hypersurfaces where the appropriate bulk probem is a singular version of the classical Yamabe problem of finding conformal metric rescalings that bring the scalar curvature to a constant value.

##### a polyhedron comparison theorem for 3-manifolds with positive scalar curvature

We establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is a isometric to a flat polyhedron.

##### Index and spectrum of minimal hypersurfaces arising from the Allen-Cahn construction

The Allen-Cahn construction is a method for constructing minimal surfaces of codimension 1 in closed manifolds. In this approach, minimal hypersurfaces arise as the weak limits of level sets of critical points of the Allen-Cahn energy functional. This talk will relate the variational

properties of the Allen-Cahn energy to those of the area functional on the surface arising in the limit, under the assumption that the limit surface has a unit normal section. In this case, bounds for the Morse indices of the critical points lead to a bound for the Morse index of the limit minimal surface. As a corollary, minimal hypersurfaces arising from an Allen-Cahn p-parameter min-max construction have index at most p. An analogous argument also establishes a lower bound for the spectrum of the Jacobi operator of the limit surface.

##### Uniqueness of blow-ups and asymptotic decay for Dirichlet energy minimizing multi-valued functions

***Please note special date and time. **In the early 1980's Almgren developed a theory of Dirichlet energy minimizing multi-valued functions, proving that the Hausdorff dimension of the singular set (including branch points) of such a function is at most (n-2) where n is the dimension of its domain. Almgren used this bound in an essential way to show that the same upper bound holds for the dimension of the singular set of an area minimizing n-dimensional rectifiable current of arbitrary codimension. In either case, the dimension bound is sharp. I will describe joint work with Brian Krummel in which we develop estimates to study the asymptotic behaviour of a Dirichlet energy minimizing q-valued function on approach to its branch set. Our estimates imply that a Dirichlet energy minimizer at a.e.