arXiv research

Compact embeddings for invariant functions in metric-measure spaces.

Trivial solution proof for heat equation on certain manifolds.

Compact metrics found with specific curvature properties on 3D surfaces.

In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We fin…

On a Riemannian compact manifold, we give existence and multiplicity results for solutions of elliptic PDE by introducing isometry invariances. When the groups we used have finite orbits, we get multiplicity results for equations with the classical critical Sobolev exponent, for instance the Yamabe equation. When there…

The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.

Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp $L^q$ regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of $\mathbb{R}^n$, which imply asymptotic behavior of the solutions at i…

We study the geodesic distance induced by right-invariant metrics on the group $\operatorname{Diff}_c(M)$ of compactly supported diffeomorphisms of a manifold $M$, and show that it vanishes for the critical Sobolev norms $W^{s,n/s}$, where $n$ is the dimension of $M$ and $s\in(0,1)$. This completes the proof that the g…

The use of certain critical-exponent Sobolev norms is an important feature of methods employed by Taubes to solve the anti-self-dual and similar non-linear elliptic partial differential equations. Indeed, the estimates one can obtain using these critical-exponent norms appear to be the best possible when one needs to b…

Study semilinear equations on weighted manifolds to prove rigidity.

We study existence, uniqueness and stability of radial solutions of the Lane-Emden-Fowler equation $-Δ_g u=|u|^{p-1}u$ in a class of Riemannian models $(M,g)$ of dimension $n\ge 3$ which includes the classical hyperbolic space $\mathbb H^n$ as well as manifolds with sectional curvatures unbounded below. Sign properties…

Let (M,g) be a compact Riemannian three-dimensional manifold with boundary. We prove the compactness of the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. This involves a blow-up analysis of a Yamabe-type equation with critical Sobolev e…

The problem of prescribing conformally the scalar curvature of a closed Riemannian manifold as a given Morse function reduces to solving an elliptic partial differential equation with critical Sobolev exponent. Two ways of attacking this problem consist in subcritical approximations or negative pseudo gradient flows. W…

Given a compact Riemannian Manifold (M,g) of dimension n > 2, a point x_0 in M and s in (0,2). We let 2*(s) = 2(n-s)/(n-2) be the critical Hardy-Sobolev exponent. The Hardy-Sobolev embedding yields the existence of A,B > 0 such that (\int_M|u|^{2*(s)}dv_g)^{2/2*(s)} \leq A\int_M |\nabla u|_g^2 dv_g +B\int_M u^2 dv_g fo…

Let (M,g) be a compact Riemannien Manifold of dimension n > 2, x_0 in M a fix and singular point and s in (0,2). We let 2*(s) = 2(n-s)/(n-2) be the critical Hardy-Sobolev exponent. we investigate the existence of positive distributional solutions u in C^0(M) to the critical equation Δ_g u + a(x) u = u^{2*(s)-1}/ d_g(x,…

In this paper we study some fourth order elliptic equation involving the critical Sobolev exponent, related to the prescription of a fourth order conformal invariant on the standard sphere. We use a topological method to prove the existence of at least a solution when the function to be prescribed is close to a constan…

These are notes on seminal work of Freed, and subsequent developments, on the curvature properties of (Sobolev Lie) groups of maps from a Riemannian manifold into a compact Lie group. We are mainly interested in critical cases which are relevant to quantum field theory. For example Freed showed that, in a necessarily q…

The paper explores variational problems on Riemannian manifolds with special foliations, proving existence results.

Let $(M^n,g),~n\ge 3$ be a noncompact complete Riemannian manifold with compact boundary and $f$ a smooth function on $\partial M$. In this paper we show that for a large class of such manifolds, there exists a metric within the conformal class of $g$ that is complete, has zero scalar curvature on $M$ and has mean curv…

New groups found with critical exponents close to but less than max.

Optimal geometric estimates for Kähler manifolds with bounded Nash entropy

New proof for certain groups in higher dimensions.

In this paper we perform a fine blow-up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere. We derive from this analysis some a priori estimates in dimension 5 and 6. On the five dimensionl sphere these a prio…

Constructs free semigroups with critical exponents close to but less than ambient groups.

Proves critical exponent for $Θ-$positive representations in discrete subgroups.

Study critical exponents in normal subgroups of higher rank Lie groups.

Paper proves existence of minimum energy solutions in 5D contact spin manifolds.

We continue our previous work studying critical exponent semilinear elliptic (and subelliptic) problems which generalize the classical Yamabe problem. In [3] the focus was on metric-measure spaces with an `almost smooth' structure, with stratified spaces furnishing the key examples. The criterion for solvability there …

Study critical exponents on hyperbolic surfaces with long boundaries using Weil-Petersson measures.

New method associates topological classes to Sobolev bundles in critical dimensions.

We consider the operator algebra generated by pseudodifferential operators on a closed smooth surface and shift operator induced by a Morse--Smale diffeomorphism of this surface. Elements in this algebra are considered as operators in the scale of Sobolev spaces and the aim of this paper is to describe how Fredholm pro…

Improved Sobolev mappings in Carnot groups with weaker assumptions.

The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.

Study shows how close functions are to optimal in Riemannian manifolds.

We study the set of critical exponents of discrete groups acting on regular trees. We prove that for every real number $δ$ between $0$ and $\frac{1}{2}\log q$, there is a discrete subgroup $Γ$ acting without inversion on a $(q+1)$-regular tree whose critical exponent is equal to $δ$. Explicit construction of edge-index…

Study calculates the elastic energy of curves on a sphere.

Study on nonlinear elliptic equations with variable exponents, proving existence and multiplicity of solutions.

We propose a new Integral Probability Metric (IPM) between distributions: the Sobolev IPM. The Sobolev IPM compares the mean discrepancy of two distributions for functions (critic) restricted to a Sobolev ball defined with respect to a dominant measure $μ$. We show that the Sobolev IPM compares two distributions in hig…

Notes on quasiregular maps between Riemannian manifolds, preserving Sobolev forms.

The paper proves rigidity for complex Kleinian groups.

In this article we introduce and investigate a new two-parameter family of knot energies $TP^{(p,q)}$ that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the integrand in the original definition of the tangent-point energies. We will firs…

Study critical exponents for L^p-cohomology of higher rank Lie groups and manifolds.

Random subgroups in hyperbolic spaces have full limit sets and bounded critical exponents.

We prove that a sequence of quasi-Fuchsian representations for which the critical exponent converges to the topological dimension of the boundary of the group (larger than 2), converges up to subsequence and conjugacy to a totally geodesic representation.

Let $Γ$ be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold $X$. We show that a normal subgroup $Γ_0$ has critical exponent equal to the critical exponent of $Γ$ if and only if $Γ/ Γ_0$ is amenable. We prove a similar result for the exponential growth rate of closed geodesics on $…

Study on extremizers for Sobolev inequality on curved manifolds.

In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp.

For any geodesic current we associated a quasi-metric space. For a subclass of geodesic currents, called filling, it defines a metric and we study the critical exponent associated to this space. We show that is is equal to the exponential growth rate of the intersection function for closed curves.