arXiv research

Study finds index of symmetry for solvable 3D Lie groups with left-invariant metrics.

We prove a general relative higher index theorem for complete manifolds with positive scalar curvature towards infinity. We apply this theorem to study Riemannian metrics of positive scalar curvature on manifolds. For every two metrics of positive scalar curvature on a closed manifold and a Galois cover of the manifold…

The paper derives index-energy estimates for Yang-Mills connections and Einstein metrics.

We consider the following generalisation of a well-known problem in Riemannian geometry: When is a smooth real-valued function s on a given compact n-dimensional manifold M (with or without boundary) the scalar curvature of some smooth pseudo-Riemannian metric of index q on M? We prove that this is the case for every s…

Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the fam…

The paper identifies and constructs metrics for specific homogeneous spaces.

Study on metrics on specific nilmanifolds, finding new examples and properties.

Notwithstanding almost forty years of efforts, the market for paintings still lacks a widely accepted price index. In this paper, we introduce a simple and intuitive metric to construct such index. Our metric is based on the price of a painting divided by its area. This formulation rests on a solid mathematical foundat…

We compute the index of the real Cauchy-Riemann operator defined in FJRW theory in case of the smooth metric. For the cylindrical metric, we study the relation between the index of the linearized operator of Witten map and weights in weighted Sobolev space.

Index difference on surfaces of genus at least 3 is non-trivial.

The paper explores index theory for Dirac operators to understand scalar curvature properties.

On any surface we give an example of a metric that contains simple closed geodesics with arbitrary high Morse index. Similarly, on any 3-manifold we give an example of a metric that contains embedded minimal tori with arbitrary high Morse index. Previously no such examples were known. We also discuss whether or not suc…

Study of symmetries in 3D Lie groups, determining index and moduli space properties.

The Gauss-Bonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the Poincaré-Einstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and the dimensions of the $L^2$-cohomology spaces as well as to carry out the heat equa…

We study the Kaehler metric given by the logarithm of a cubic form on its complexified index cone. Under mirror symmetry, this metric should asymptotically correspond to the Weil-Petersson metric. Using the theory of special Kaehler manifolds, a proof of a curvature formula for this metric is given.

We determine the index of symmetry of 3-dimensional unimodular Lie groups with a left-invariant metric. In particular, we prove that every 3-dimensional unimodular Lie group admits a left-invariant metric with positive index of symmetry. We also study the geometry of the quotients by the so-called foliation of symmetry…

Deep single-index Fréchet regression for metric space-valued outputs

Given two metrics of positive scalar curvature metrics on a closed spin manifold, there is a secondary index invariant in real $K$-theory. There exist two definitions of this invariant, one of homotopical flavour, the other one defined by a index problem of Atiyah-Patodi-Singer type. We give a complete and detailed pro…

An index theorem for the anti-self-dual deformation complex on anti-self-dual orbifolds with singularities conjugate to ADE-type is proved. In 1988, Claude Lebrun gave examples of scalar-flat Kähler ALE metrics with negative mass, on the total space of the bundle $\mathcal{O}(-n)$ over $S^2$. A corollary of this index …

The paper equates the index of a vector bundle to the manifold's index and introduces an analytic torsion.

The paper proves bounds on the Morse index of free boundary minimal hypersurfaces.

Recently, Atiyah and LeBrun proved versions of the Gauss-Bonnet and Hirzebruch signature Theorems for metrics with edge-cone singularities in dimension four, which they applied to obtain an inequality of Hitchin-Thorpe type for Einstein edge-cone metrics. Interestingly, many natural examples of edge-cone metrics in dim…

New metrics contradicting old conjectures on umbilic points.

Defines new Roe algebras for cylindrical spaces, solving metric curvature problems.

This paper extends the single index model to handle nonlinear relationships.

DFR models dynamic distributional data with weighted Fréchet means.

Establishes a lower bound for Kähler-Einstein distance on certain domains.

Connected space of Dirac-minimal metrics in 2 and 4 dimensions.

Study shows infinitely many nonnegatively curved metric spaces on exotic 7-manifolds.

New Morse theory techniques glue nontransverse flowlines.

Survey on metrics and assembly maps in positive scalar curvature.

Local index theorem for cofinite hyperbolic Riemann surfaces derived from computational perspective.

The study shows how nonnegative Ricci curvature and metric cones imply the existence of abelian subgroups in the fundamental group of open manifolds.

Paper shows zero Rosenberg index for certain foliated manifolds.

We introduce partial secondary invariants associated to complete Riemannian metrics which have uniformly positive scalar curvature outside a prescribed subset on a spin manifold. These can be used to distinguish such Riemannian metrics up to concordance relative to the prescribed subset. We exhibit a general external p…

We use an index-theoretic technique of Hitchin to show that the space of complete Riemannian metrics of nonnegative sectional curvature on certain open spin manifolds has nontrivial homotopy groups in infinitely many degrees. A new ingredient of independent interest is homotopy density of the subspace of metrics with c…

We find the index of $\widetilde{\nabla}$-quasi-conformally symmetric and $\widetilde{\nabla}$-concircularly symmetric semi-Riemannian manifolds, where $\widetilde{\nabla}$ is metric connection.

In this paper we discuss the index problem for geometric differential operators (Spin-Dirac operator, Gauß-Bonnet operator, Signature operator) on manifolds with metric horns. On singular manifolds these operators in general do not have unique closed extensions. But there always exist two extremal extensions $D_{min}$ …

We construct a smooth Riemannian metric on any 3-manifold with the property that there are genus zero embedded minimal surfaces of arbitrarily high Morse index.

Study of psc metrics via block diffeomorphisms and cubical sets.

Study free boundary minimal submanifolds in geodesic balls in hyperbolic and spherical spaces.

Proves Morse index theorem for geodesics in conic Finsler manifolds.

Let $(M^{n+1},g,e^{-f}dμ)$ be a complete smooth metric measure space with $2\leq n\leq 6$ and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded $f$-minimal hypersurfaces in $M$ with uniform upper bounds on $f$-index and weighted vo…

Proves a strengthening of Chang, Weinberger, and Yu's theorem on positive scalar curvature.

Study on geodesics proving index and intersection bounds, with examples of multiplicity.

Extends index theorem to domain walls with discontinuous Riemannian connections.

The paper proposes methods to directly optimize complex classification metrics.

The paper provides obstructions to positive scalar curvature for certain manifolds with group actions.