The Webster scalar curvature flow on CR sphere. Part II

This is the second of two papers, in which we study the problem of prescribing Webster scalar curvature on the CR sphere as a given function f. Using the Webster scalar curvature flow, we prove an existence result under suitable assumptions on the Morse indices of f.     

A gap theorem on submanifolds with finite total curvature in spheres

We study a complete noncompact submanifold Mⁿ$M^n$ in a sphere S^n+p${\mathbb{S}}^{n+p}$. We prove that there admit no nontrivial L²$L^2$-harmonic 1-forms on M if the total curvature is bounded from above by a constant depending only on n. The gap theorem is a generalized version of Carron?s, Yun?s, Cavalcante?s and the first author?s results on submanifolds in Euclidean spaces and Seo?s result on submanifolds in hyperbolic space without the condition of minimality.     

Isoperimetric type problems and Alexandrov–Fenchel type inequalities in the hyperbolic space

In this paper, we solve various isoperimetric problems for the quermassintegrals and the curvature integrals in the hyperbolic space Hⁿ${\mathbb{H}}^n$, by using quermassintegral preserving curvature flows. As a byproduct, we obtain hyperbolic Alexandrov–Fenchel inequalities.     

The Webster scalar curvature problem on the three dimensional CR manifolds

In this paper, we prove some existence results for the Webster scalar curvature problem on the three dimensional CR compact manifolds locally conformally CR equivalent to the unit sphere S³ of C². Our methods are based on the techniques related to the theory of critical points at infinity.     

Dirichlet operators: A priori estimates and uniqueness problems II

We study some basic properties of the Dirichlet operator in weighted L^p$L^p$ spaces. We work under rather general assumptions on weights that destroy minimal local Sobolev regularity of the form domain.     

Hardy and uncertainty inequalities on stratified Lie groups

We prove various Hardy-type and uncertainty inequalities on a stratified Lie group G  . In particular, we show that the operators _Tα:f?|⋅|^−αL^−α/2f$T_{\alpha }:f?{|\cdot |}^{-\alpha }{L}^{-\alpha /2}f$, where |⋅|$|\cdot |$ is a homogeneous norm, 0<α<Q/p$0<\alpha <Q/p$, and L   is the sub-Laplacian, are bounded on the Lebesgue space L^p(G)$L^p(G)$. As consequences, we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality. We also deduce a general version of the Heisenberg–Pauli–Weyl inequality, relating the L^p$L^p$ norm of a function f   to the L^q$L^q$ norm of |⋅|^βf${|\cdot |}^{\beta }f$ and the L^r$L^r$ norm of L^δ/2f$L^{\delta /2}f$.     

Bertrand curves of AW(k)-type in Lorentzian space

In this study, firstly, we give curvature conditions of AW(k)-type (1≤k≤3$1\le k\le 3$) curves. Then considering AW(k)-type curves, we investigate Bertrand curves γ:I→L³$\gamma :I\to L^3$ with _κ1(s)≠0${\kappa }_1\left(s\right)\ne 0$ and _κ2(s)≠0${\kappa }_2\left(s\right)\ne 0$. We show that there are Bertrand curves of AW(1)-type and AW(3)-type. Moreover, we study weak AW(2)-type and AW(3)-type conical geodesic curves in L³$L^3$.     

The Calderón–Zygmund inequality and Sobolev spaces on noncompact Riemannian manifolds

We introduce the concept of Calderón–Zygmund inequalities on Riemannian manifolds. For 1<p<∞$1<p<\infty$, these are inequalities of the form

_{‖Hess(u)‖Lp}≤_C1‖u‖Lp+_{C2‖Δu‖Lp},${\Vert \mathrm{Hess}(u)\Vert }_{{\mathsf{L}}^p}\le C_1{\Vert u\Vert }_{{\mathsf{L}}^p}+C_2{\Vert \mathrm{\Delta }u\Vert }_{{\mathsf{L}}^p},$

valid a priori for all smooth functions u   with compact support, and constants _C1≥0$C_1\ge 0$, _C2>0$C_2>0$. Such an inequality can hold or fail, depending on the underlying Riemannian geometry. After establishing some generally valid facts and consequences of the Calderón–Zygmund inequality (like new denseness results for second order L^p${\mathsf{L}}^p$-Sobolev spaces and gradient estimates), we establish sufficient geometric criteria for the validity of these inequalities on possibly noncompact Riemannian manifolds. These results in particular apply to many noncompact hypersurfaces of constant mean curvature.     

A transfer integral technique for solving a class of linear integral equations: Convergence and applications to DNA

An eigenvalue problem, the convergence difficulties that arise and a mathematical solution are considered. The eigenvalue problem is motivated by simplified models for the dissociation equilibrium between double-stranded and single-stranded DNA chains induced by temperature (thermal denaturation), and by the application of the so-called transfer integral technique. Namely, we extend the Peyrard–Bishop model for DNA melting from the original one-dimensional model to a three-dimensional one, which gives rise to an eigenvalue problem defined by a linear integral equation whose kernel is not in L²$L^2$. For the one-dimensional model, the corresponding kernel is not in L²$L^2$ either, which is related to certain convergence difficulties noticed by previous researchers. Inspired by methods from quantum scattering theory, we transform the three-dimensional eigenvalue problem, obtaining a new L²$L^2$ kernel which has improved convergence properties.     

The natural Banach space for version independent risk measures

Risk measures, or coherent measures of risk, are often considered on the space L^∞$L^{\infty }$, and important theorems on risk measures build on that space. Other risk measures, among them the most important risk measure–the Average Value-at-Risk–are well defined on the larger space L¹$L^1$ and this seems to be the natural domain space for this risk measure. Spectral risk measures constitute a further class of risk measures of central importance, and they are often considered on some L^p$L^p$ space. But in many situations this is possibly unnatural, because any L^p$L^p$ with p>_p0$p>p_0$, say, is suitable to define the spectral risk measure as well. In addition to that, risk measures have also been considered on Orlicz and Zygmund spaces. So it remains for discussion and clarification, what the natural domain to consider a risk measure is?     

The cutoff phenomenon for Ehrenfest chains

We consider families of Ehrenfest chains and provide a simple criterion on the L^p$L^p$-cutoff and the L^p$L^p$-precutoff with specified initial states for 1≤p<∞$1\le p<\infty$. For the family with an L^p$L^p$-cutoff, a cutoff time is described and a possible window is given. For the family without an L^p$L^p$-precutoff, the exact order of the L^p$L^p$-mixing time is determined. The result is consistent with the well-known conjecture on cutoffs of Markov chains proposed by Peres in 2004, which says that a cutoff exists if and only if the multiplication of the spectral gap and the mixing time tends to infinity.     

Hamiltonian-minimal Lagrangian submanifolds in Kaehler manifolds with symmetries

By making use of the symplectic reduction and the cohomogeneity method, we give a general method for constructing Hamiltonian minimal Lagrangian submanifolds in Kaehler manifolds with symmetries. As applications, we construct infinitely many nontrivial complete Hamiltonian minimal Lagrangian submanifolds in CPⁿ$C{P}^n$ and Cⁿ$C^n$.     

Integral geometry for the 1-norm

Classical integral geometry takes place in Euclidean space, but one can attempt to imitate it in any other metric space. In particular, one can attempt this in Rⁿ${\mathbb{R}}^n$ equipped with the metric derived from the p  -norm. This has, in effect, been investigated intensively for 1<p<∞$1<p<\infty$, but not for p=1$p=1$. We show that integral geometry for the 1-norm bears a striking resemblance to integral geometry for the 2-norm, but is radically different from that for all other values of p  . We prove a Hadwiger-type theorem for Rⁿ${\mathbb{R}}^n$ with the 1-norm, and analogues of the classical formulas of Steiner, Crofton and Kubota. We also prove principal and higher kinematic formulas. Each of these results is closely analogous to its Euclidean counterpart, yet the proofs are quite different.     

Harness processes and harmonic crystals

In the Hammersley harness processes the R$\mathbb{R}$-valued height at each site i∈Z^d$i\in {\mathbb{Z}}^d$ is updated at rate 1 to an average of the neighboring heights plus a centered random variable (the noise). We construct the process “a la Harris” simultaneously for all times and boxes contained in Z^d${\mathbb{Z}}^d$. With this representation we compute covariances and show L²$L^2$ and almost sure time and space convergence of the process. In particular, the process started from the flat configuration and viewed from the height at the origin converges to an invariant measure. In dimension three and higher, the process itself converges to an invariant measure in L²$L^2$ at speed t^1−d/2$t^{1-d/2}$ (this extends the convergence established by Hsiao). When the noise is Gaussian the limiting measures are Gaussian fields (harmonic crystals) and are also reversible for the process.     

Stability estimates for the Radon transform with restricted data and applications

In this article, we prove a stability estimate going from the Radon transform of a function with limited angle-distance data to the L^p$L^p$ norm of the function itself, under some conditions on the support of the function. We apply this theorem to obtain stability estimates for an inverse boundary value problem with partial data.     

Estimates for eigenvalues of the Paneitz operator

For an n  -dimensional compact submanifold Mⁿ$M^n$ in the Euclidean space R^N${\mathbf{R}}^N$, we study estimates for eigenvalues of the Paneitz operator on Mⁿ$M^n$. Our estimates for eigenvalues are sharp.     

Insensitizing controls with one vanishing component for the Navier–Stokes system

In this paper we prove the existence of insensitizing controls, having one vanishing component, for the local L²$L^2$-norm of the solutions of the Navier–Stokes system. This problem can be recast as a null controllability problem for a nonlinear cascade system. We first prove a controllability result, with controls having one vanishing component, for a linear problem. Then, by means of an inverse mapping theorem, we deduce the controllability for the cascade system.