arXiv research

Unique cylindrical tangent cone for Simons' hypersurface found.

Uniqueness of nondegenerate blowups for planar networks shown.

Uniqueness proven for cylindrical tangent cones in high dimensions.

This paper solves minimal surface equations near Hardt-Simon foliations.

A note on the uniqueness of differential characters and K-theory via homological algebra.

W. Simon proved a conformal positive mass theorem, which was used to prove uniqueness of black holes later. In this note, we will generalize Simon's conformal positive mass theorem in two directions. First we will consider spacetime version of conformal positive mass theorems on asymptotically flat initial data set. Ne…

Gradient flow of elastic energy converges to elastica.

New resurgent analysis reveals dual $q$-series for Chern-Simons theory crossing natural boundaries.

The study examines singularities in flows with curvature bounds and identifies unique tangent flows.

We refine a Le and Murakami uniqueness theorem for the Kontsevich Integral in order to specify the relationship between the two (possibly equal) main universal link invariants: the Kontsevich Integral and the perturbative expression of the Chern-Simons theory. As a corollary, we prove that the Altschuler and Freidel an…

Paper formalizes Simon's satisficing through FFSD, proving its equivalence to expected utility theory.

New approach reveals causal and probabilistic relationships from equations.

New proof of harmonic map uniqueness with analytic targets.

In this paper we give explicit formulas of differential characteristic classes of principal $G$-bundles with connections and prove their expected properties. In particular, we obtain explicit formulas for differential Chern classes, differential Pontryagin classes and differential Euler class. Furthermore, we show that…

The paper proves uniqueness of solutions to curvature problems using various methods.

Paper confirms Thom's conjecture for nonlinear evolutions on manifolds.

We study Cheeger-Simons differential characters and provide geometric descriptions of the ring structure and of the fiber integration map. The uniqueness of differential cohomology (up to unique natural transformation) is proved by deriving an explicit formula for any natural transformation between a differential cohom…

Complex Chern-Simons theory reveals peacock patterns in perturbative series.

The paper proves new inequalities and flow properties for hypersurfaces.

We prove the local existence of unique smooth solutions of the Donaldson geometric flow on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. It is a semiflow on the Besov space $B^{1,p}_2(M, Λ^2)$ for $p > 4$. The Donaldson geometric flow was introduced by Simon Dona…

Construct locally minimizing $(1,2)$-clusters with prescribed asymptotic geometry.

Odd $K$-theory has the interesting property that it admits an infinite number of inequivalent differential refinements. In this paper we provide a bundle theoretic model for odd differential $K$-theory using the caloron correspondence and prove that this refinement is unique up to a unique natural isomorphism. We chara…

The paper proves smooth convergence of evolving hypersurfaces to critical points.

We introduce certain relative differential characters which we call Cheeger-Chern-Simons characters. These combine the well-known Cheeger-Simons characters with Chern-Simons forms. In the same way as the Cheeger-Simons characters generalize Chern-Simons invariants of oriented closed manifolds, the Cheeger-Chern-Simons …

Study on unique solutions to one-phase free boundary problems.

We prove the two theorems of the title, settling two long standing questions in the local theory of singular minimal hypersurfaces. The sharpness of either result is with respect to its hypothesis on the size of the allowable singular sets. The proofs of both theorems rely heavily on the author's recent regularity and …

Computing Chern-Simons action for perturbed Dirac triples

In the asymptotically locally hyperbolic setting it is possible to have metrics with scalar curvature at least -6 and negative mass when the genus of the conformal boundary at infinity is positive. Using inverse mean curvature flow, we prove a Penrose inequality for these negative mass metrics. The motivation comes fro…

New approach connects 3D Chern-Simons theory to spectral networks.

We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,α}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…

We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,α}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…

Alternative proof of Michael-Simon-Sobolev inequality using optimal transport.

Geometrically constructs dilogarithm from Chern-Simons theory.

We define the Simons-Sullivan differential analytic index by translating the Freed-Lott differential analytic index via explicit ring isomorphisms between Freed-Lott differential K-theory and Simons-Sullivan differential K-theory. We prove the differential Grothendieck-Riemann-Roch theorem in Simons-Sullivan differenti…

Arithmetic Dijkgraaf-Witten theory constructs analogues in Chern-Simons TQFT.

We extend the interior gradient estimate due to N. Korevaar and L. Simon for solutions of the mean curvature equation from the case of Euclidean graphs to the general case of Killing graphs. Our main application is the proof of existence of Killing graphs with prescribed mean curvature function for continuous boundary …

The abstract discusses conjectures about Chern-Simons invariants of 3-manifolds.

We calculate the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.

New $L_\infty$ liftings derived from Chern-Simons classes for coherent sheaves.

The paper explores the connection between 3d gravity and Chern-Simons theory using affine group connections.

We show, for mean curvature flows in Euclidean space, that if one of the tangent flows at a given space-time point consists of a closed, multiplicity-one, smoothly embedded self-similar shrinker, then it is the unique tangent flow at that point. That is the limit of the parabolic rescalings does not depend on the chose…

Lecture notes on Lie groups and Chern-Simons theory for grad students.

We apply our abstract gradient inequalities developed by the authors in arXiv:1510.03817 to prove Lojasiewicz--Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Our Lojasiewicz--Simon gradient …

We prove several abstract versions of the Lojasiewicz-Simon gradient inequality for an analytic functional on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, the well-known infinite-dimensional version of the gradient inequality due to Lojasie…

Develops a TQFT framework to compute $\hat{Z}$ invariants of three-manifolds.

Chern-Simons and Reshetikhin-Turaev theories are shown equivalent for U(1) gauge group.

We prove Simon's conjecture for 3-manifolds.

We calculate the Chern-Simons invariants of the hyperbolic double twist knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of cone-manifold structures of double twist knots.