arXiv research

Sharp bounds on scalar curvature spectrum and rigidity theorems.

Study pinches curvature under Laplacian G_2 flow, proving Weyl tensor norm blows up.

Investigate scalar curvature under geometric flows

New examples show positive scalar curvature metrics on manifolds with boundary that cannot be extended.

Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.

Estimates for scalar curvature equations on Kähler manifolds with singularities.

Paper gives curvature estimates for Laplacian flow on G_2-structures.

Optimal controls for conformal Laplacian obstacle problems on spheres and manifolds.

Formula for Lichnerowicz Laplacian on invariant metrics, deducing stability of Einstein manifolds.

New definitions and properties of harmonic vector fields on Finsler manifolds.

The study finds bounds for the first eigenvalue of the p-Laplacian on submanifolds.

Paper extends noncompact Llarull's theorem to manifolds with boundary.

Simplified Obata-Vétois argument for Einstein manifolds with nonnegative scalar curvature.

Study stability of Einstein metrics on homogeneous spaces.

Researchers calculate the precise boundary operator for interacting bulk scalar fields in AdS/CFT.

We show the analytic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces on differential forms, including high energy estimates in strips. This is achieved by placing the spectral family of the Laplacian within the framework developed, and applied to scalar problems, by the author recentl…

In this paper we study the curved geometry of noncommutative 4-tori $\mathbb{T}_θ^4$. We use a Weyl conformal factor to perturb the standard volume form and obtain the Laplacian that encodes the local geometric information. We use Connes' pseudodifferential calculus to explicitly compute the terms in the small time hea…

A new derivation is given of Branson's factorization formula for the conformally invariant operator on the sphere whose principal part is the k-th power of the scalar Laplacian. The derivation deduces Branson's formula from knowledge of the corresponding conformally invariant operator on Euclidean space (the k-th power…

Gradient η-Ricci solitons on warped products are studied.

Study pseudo-laplacians and ζ(1) for spinor bundles over Riemann surfaces.

New eigenvalue bounds for 3-Sasaki metrics improve previous estimates.

The paper studies a flow of hypersymplectic structures linked to $G_2$-geometry.

For a Riemannian closed spin manifold and under some topological assumption (non-zero $\hat{A}$-genus or enlargeability in the sense of Gromov-Lawson), we give an optimal upper bound for the infimum of the scalar curvature in terms of the first eigenvalue of the Laplacian. The main difficulty lies in the study of the o…

We study the spectral geometric properties of the scalar Laplace-Beltrami operator associated to the Weil-Petersson metric $g_{\mathrm{WP}}$ on $\mathcal M_γ$, the Riemann moduli space of surfaces of genus $γ> 1$. This space has a singular compactification with respect to $g_{\mathrm{WP}}$, and this metric has crossing…

The paper proves estimates for Hodge Laplacians on Lie groups.

New formula for Lichnerowicz Laplacian on homogeneous spaces.

For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which in…

The Yamabe flow affects the first eigenvalues of geometric operators on manifolds.

The paper bounds eigenvalues of magnetic Schroedinger operators on compact manifolds.

Study stability of operators on warped product manifolds.

In a noncommutative torus, effect of perturbation by inner derivation on the associated quantum stochastic process and geometric parameters like volume and scalar curvature have been studied. Cohomological calculations show that the above perturbation produces new spectral triples. Also for the Weyl C^*-algebra, the La…

A simple formula is derived for the Ricci scalar curvature of any smooth level set ${ψ(x_0,x_1,...,x_n)=C}$ embedded in the Euclidean space $\mathbb R^{n+1}$, in terms of the gradient $\nablaψ$ and the Laplacian $Δψ$. Some applications are given to the geometry of low-dimensional $p$-harmonic functions and high-dime…

We show that, on any asymptotically hyperbolic surface, the essential spectrum of the Lichnerowicz Laplacian $Δ_L$ contains the ray $[{1/4},+\infty[$. If moreover the scalar curvature is constant then -2 and 0 are infinite dimensional eigenvalues. If, in addition, the inequality $<Δu, u>_{L^2}\geq \frac14||u||^2_{L^2}$…

Derives inequalities for eigenvalues and renormalized volume of Poincaré-Einstein manifolds.

We study the heat kernel asymptotics for the Laplace type differential operators on vector bundles over Riemannian manifolds. In particular this includes the case of the Laplacians acting on differential p-forms. We extend our results obtained earlier for the scalar Laplacian and present closed formulas for all heat in…

The paper solves conditions for prescribing scalar and Gauss curvatures on manifolds with zero first eigenvalue.

Proves infinite bordism groups for certain manifolds with positive scalar curvature.

We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yam…

The paper connects curvature data to polynomial coefficients in gluing formulas.

Prove that collapsing CSC metrics can be perturbed to invariant collapsing CSC metrics.

We prove uniqueness results for a Calderon type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. T…

Estimates eigenvalues using Bessel functions on manifolds.

Inverse scattering result on AH manifolds determines metric up to diffeo and conformal factor.

Paper studies constant scalar curvature equation and its smooth solutions on Kähler manifolds.

For the dual operator $s_g'^*$ of the linearization $s_g'$ of the scalar curvature function, it is well-known that if $\ker s_g'^*\neq 0$, then $s_g$ is a non-negative constant. In particular, if the Ricci curvature is not flat, then ${s_g}/(n-1)$ is an eigenvalue of the Laplacian of the metric $g$. In this work, some…

Study shows obstructions to positive scalar curvature cobordisms using periodic η-invariants.

The paper develops a new Laplacian for manifold learning from data.

Study shows equivalence of two methods for solving scalar curvature problem.