 # Geometry and Analysis on Manifolds 2018 University of California, Santa Barbara May 4-6, 2018

### Date, Time and Location:

The Geometry and Analysis on Manifold conference (GAMSB18) will be held at the Department of Mathematics of University of California at Santa Barbara From May 4, 2018 to May 6, 2018. The lectures will be held in South Hall 6635. For directions on how to get to South Hall 6635, here is a map.

The conference is supported by the Research Training Groups (RTG) in Topology and Geometry funded by the Division of Mathematical Sciences (DMS) of the National Science Foundation (NSF) and College of Letter and Science, University of California at Santa Barbara.

### Confirmed Speakers:

• Chen, Lina   (East China Normal University)
• Fraser, Ailana   (University of British Columbia)
• Graham, Robin   (University of Washington)
• Gursky, Matt   (Notre Dame University)
• Haslhofer, Robert   (University of Toronto)
• Li, Haizhong   (Tsinghua University)
• Mantoulidis, Christos   (MIT)
• Mazzeo, Rafe   (Stanford University)
• Rong, Xiaochun   (Rutgers University)
• Searle, Catherine   (Wichita State University)
• Varghese, Mathai   (University of Adelaide)
• Wang, Hang   (University of Adelaide)
• Wang, McKenzie   (McMaster University)
• Zhang, Weiping   (Chern Institute)
• Zhou, Detang   (Universidade Federal Fluminense)

### Abstracts:

• Chen, Lina: A geometric approach to Milnor's problem on group growth

Abstract: The modified Milnor's problem on group growth asks that whether any finitely presented group of vanishing algebraic entropy has at most polynomial growth. We will show that a positive answer to the modified Milnor's problem is equivalent to the following nilpotency conjecture: Given any $$n>0, d>0$$, there is $$\epsilon(n, d)>0$$ such that if a $$n$$-manifold $$M$$ satisfies $$Ric\geq -(n-1), \textit{diam}(M)\leq d$$ then the fundamental group of $$M$$ is virtually nilpotent or the volume entropy of $$M$$ is not smaller than $$\epsilon(n, d)$$. We will also verify several cases for a gap phenomena of volume entropy related to the nilpotency conjecture. This is a joint work with Xiaochun Rong and Shicheng Xu.

• Fraser, Ailana: Shape optimization for an eigenvalue problem on manifolds with boundary

Abstract: When we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases it makes sense to attempt to find critical points in the space of metrics. For surfaces, the critical metrics turn out to be the induced metrics on certain special classes of minimal (mean curvature zero) surfaces in spheres and Euclidean balls. The eigenvalue extremal problem is thus related to other questions arising in the theory of minimal surfaces. In this talk we will give an overview of progress that has been made for surfaces with boundary, and contrast this with some recent results in higher dimensions. This is joint work with R. Schoen.

• Graham, Robin: Geodesic flow, X-ray transform, and boundary rigidity on asymptotically hyperbolic manifolds

Abstract: I will describe an extension to the boundary of the cosphere bundle and geodesic flow of an asymptotically hyperbolic manifold. I will then discuss injectivity results for X-ray transforms on tensors and the boundary rigidity problem of determining an asymptotically hyperbolic metric from the renormalized lengths of geodesics joining boundary points. This is joint work with Colin Guillarmou, Plamen Stefanov and Gunther Uhlmann.

• Gursky, Matt: Obstructions to the existence of conformally compact Einstein manifolds

Abstract: In this talk I will describe a singular boundary value problem for Einstein metrics. This problem arises in the Fefferman-Graham theory of conformal invariants, and in the AdS/CFT correspondence. After giving a brief overview of some important results and examples, I will present a recent construction of boundary data which cannot admit a solution. Finally, I will introduce a more general index-theoretic invariant which gives an obstruction to existence in the case of spin manifolds. This is joint work with Q. Han and S. Stolz.

• Haslhofer, Robert: Minimal two-spheres in three-spheres

Abstract: We prove that any manifold diffeomorphic to S^3 and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in 1983. Our approach combines ideas from min-max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in three-manifolds. Finally, we apply our methods to solve a problem posed by S.T. Yau in 1987, and to show that the assumptions in the multiplicity one conjecture and the equidistribution of widths conjecture are in a certain sense sharp. This is joint work with Dan Ketover.

• Li, Haizhong: Variational problems for Riemannian functionals and the Generalized Willmore conjecture

Abstract: In this talk, we present our research works about variational problems of Riemannian functionals on an n-dimensional compact Riemmannian manifold $$(M,g)$$, which include the renormalized volume coefficients functional $$\int_M v^{2k}(g)dv_g$$, and the Weyl curvature functional $$\int_M |W(g)|^{n/2}dv$$. For a hypersurface in a sphere, we study the generalized Willmore functional and generalized Willmore conjecture. By use of an inequality between the Weyl curvature functional and the generalized Willmore functional, we give some discussions about the Generalized Willmore conjecture for 4-dimensional compact hypersurfaces in a sphere.

• Mantoulidis, Christos: Minimal surfaces and the Allen-Cahn equation on 3-manifolds

Abstract: This is joint work with O. Chodosh. The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. Using new curvature estimates and sharp sheet separation estimates for stable Allen-Cahn solutions on 3-manifolds, derived by improving recent work of Wang-Wei, we show: minimal surfaces arising from Allen-Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index, provided the ambient metric is generic. This confirms, in the Allen-Cahn setting, two conjectures of Marques-Neves: a strengthened multiplicity one conjecture, and the index lower bound conjecture. If combined with recent work of Guaraco and Gaspar-Guaraco, this gives a new proof of Yau's conjecture on infinitely many minimal surfaces in a 3-manifold with a generic metric (recently proven in dimensions up to 7 by Irie-Marques-Neves) with new geometric conclusions.

• Mazzeo, Rafe: Some new aspects in gauge theory

Abstract: A conjecture by Gaiotto and Witten proposes a gauge-theoretic interpretation of the Jones polynomials and Khovanov homology. This relies on an analysis of the Kapustin-Witten equations. This talk will describe recent progress on several analytic aspects of this program. Joint work with Witten and with He.

• Rong, Xiaochun: Collapsed manifolds with Ricci local bounded covering geometry

Abstract: Collapsed manifolds with local bounded covering geometry (i.e., sectional curvature bounded in absolute value) has been well-studied; the basic discovery by Cheeger-Fukaya-Gromov is the existence of a compatible local nilpotent symmetry structures whose orbits point to all collapsed directions. In this talk, we will report on generalizing the structural result to collapsed manifolds with (partially) local Ricci bounded covering geometry. Our construction of local nilpotent symmetry structures does not reply on the work of Cheeger-Fukaya-Gromov; which gives alternative approach to the structural result.

• Searle, Catherine: 4-dimensional positively curved regular Alexandrov spaces with circle symmetry

Abstract: We classify positively curved regular Alexandrov spaces (including Riemannian orbifolds) of dimension 4 admitting an isometric circle action. This is joint work with John Harvey.

• Varghese, Mathai: Positive scalar curvature via end-periodic manifolds and generalized eta invariants

Abstract: I will discuss some research with my MPhil student, Michael Hallam.

• Wang, Hang: Higher Nahm transform in noncommutative geometry

Abstract: Anti-self-dual (ASD) connections for a compact smooth four manifold arise as critical values for the Yang-Mills action functional. Nahm transform is a nice correspondence between a vector bundle with ASD connections and vector bundle with ASD connections over Picard torus associated to X. In this talk we propose a noncommutative geometric version of the Nahm transform that generalises the Connes-Yang-Mills action functional formulated using Dixmier trace. This is joint work with Tsuyoshi Kato and Hirofumi Sasahira.

• Wang, McKenzie: Examples of Ancient Ricci Flows on Bundles

Abstract: I will report on joint work with Peng Lu (U. of Oregon) in which we construct families of ancient solutions of type I of the Ricci flow on certain torus and RP^3 bundles and identify their forwards and backwards time-rescaled limits. These examples generalize those of Bakas, Kong and Ni.

• Zhang, Weiping: Positive scalar curvature and the Euler class

Abstract: We will describe a joint work with Jianqing Yu which extends the classical Lichnerowicz vanishing theorem to a twisted case involving the Euler class of a flat vector bundle.

• Zhou, Detang: Eigenvalues on Self-expanders of mean curvature flows

Abstract: I will discuss the eigenvalues of the corresponding drifted Laplacian on self-expanders of mean curvature. We show the discreteness of the spectrum and give a universal lower bound of the bottom of the spectrum of the drifted Laplacian. This lower bound is achieved if and only if the self-expander is the Euclidean subspace through the origin. For self-expanders of codimension 1, we prove an inequality between the bottom of the spectrum of the drifted Laplacian and the bottom of the spectrum of weighted stability operator and that the hyperplane through the origin is the unique self-expander where the equality holds. This is a joint work with X. Cheng.

### Hotels:

We recommend the following hotels. Please mention the Conference and the UCSB Department of Mathematics at the time of making the reservation.

The hotels have shuttles to campus.

### Organizing Committee:

Questions? e-mail to: zhou@math.ucsb.edu