Renormalization of potentials and generalized centers

We generalize the Riesz potential of a compact domain in Rm${\mathbb{R}}^m$ by introducing a renormalization of the rα−m$r^{\alpha -m}$-potential for α?0$\alpha ?0$. This can be considered as generalization of the dual mixed volumes of convex bodies as introduced by Lutwak. We then study the points where the extreme values of the (renormalized) potentials are attained. These points can be considered as a generalization of the center of mass. We also show that only balls give extreme values among bodied with the same volume.     

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.     

Partial regularity results up to the boundary for harmonic maps into a Finsler manifold

We study the energy functional for maps from a Riemannian m-manifold M   into a Finsler space N=(Rn,F)$N=({\mathbb{R}}^n,F)$. Under the restriction 2?m?4$2?m?4$, we prove the full Hölder regularity of weakly harmonic maps (i.e., weak solutions of its Euler–Lagrange equation) from M to N   in the case that the Finsler structure F(u,X)$F(u,X)$ depends only on vectors X, and a partial Hölder regularity of energy minimizing maps in general cases.     

Absence of robust rigidity for circle maps with breaks

We give examples of analytic circle maps with singularities of break type with the same rotation number and the same size of the break for which no conjugacy is Lipschitz continuous. In the second part of the paper, we discuss a class of rotation numbers for which a conjugacy is C¹$C^1$-smooth, although the numbers can be strongly non-Diophantine (Liouville). For the rotation numbers in this class, we construct examples of analytic circle maps with breaks, for which the conjugacy is not C^1+α$C^{1+\alpha }$ smooth, for any α>0$\alpha >0$.     

Homotopy classes of harmonic maps of the stratified 2-spheres and applications to geometric flows

We show that the set of harmonic maps from the 2-dimensional stratified spheres with uniformly bounded energies contains only finitely many homotopy classes. We apply this result to construct infinitely many harmonic map flows and mean curvature flows of 2-sphere in the connected sum of two closed 3-dimensional manifolds _M1≠S³$M_1\ne S^3$ and _M2≠S³,RP³$M_2\ne S^3,\mathbb{R}{\mathbb{P}}^3$, which must develop finite time singularity.     

Classification of invariant valuations on the quaternionic plane

We describe the orbit space of the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$ on the real Grassmann manifolds _Grk(H²)${\mathrm{Gr}}_k({\mathbb{H}}^2)$ in terms of certain quaternionic matrices of Moore rank not larger than 2. We then give a complete classification of valuations on the quaternionic plane H²${\mathbb{H}}^2$ which are invariant under the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$.     

BOCHNER TECHNIQUE ON STRONG KHLER-FINSLER MANIFOLDS

By using the Chern-Finsler connection and complex Finsler metric,the Bochner technique on strong Khler-Finsler manifolds is studied.For a strong Khler-Finsler manifold M,the authors first prove that there exists a system of local coordinate which is normalized at a point v ∈ M-=T 1,0M\o(M),and then the horizontal Laplace operator H for diffierential forms on PTM is defined by the horizontal part of the Chern-Finsler connection and its curvature tensor,and the horizontal Laplace operator H on holomorphic vector bundle over PTM is also defined.Finally,we get a Bochner vanishing theorem for diffierential forms on PTM.Moreover,the Bochner vanishing theorem on a holomorphic line bundle over PTM is also obtained     

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$L^{\delta /2}f$.     

Local and global properties of solutions of quasilinear Hamilton–Jacobi equations

We study some properties of the solutions of (E) −_Δpu+|∇u|^q=0$-{\mathrm{\Delta }}_{p}u+{|\mathrm{\nabla }u|}^q=0$ in a domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, mostly when p≥q>p−1$p\ge q>p-1$. We give a universal a priori estimate of the gradient of the solutions with respect to the distance to the boundary. We give a full classification of the isolated singularities of the nonnegative solutions of (E), a partial classification of isolated singularities of the negative solutions. We prove a general removability result expressed in terms of some Bessel capacity of the removable set. We extend our estimates to equations on complete noncompact manifolds satisfying a lower bound estimate on the Ricci curvature, and derive some Liouville type theorems.     

Some Liouville-type theorems for harmonic functions on Finsler manifolds

In this paper, we give some Liouville-type theorems for L^p$L^p$(p∈R)$(p\in \mathbb{R})$ harmonic (resp. subharmonic, superharmonic) functions on forward complete Finsler manifolds. Moreover, we derive a gradient estimate for harmonic functions on a closed Finsler manifold. As an application, one obtains that any harmonic function on a closed Finsler manifold with nonnegative weighted Ricci curvature _RicN${\mathit{Ric}}_N$(N∈(n,∞))$(N\in (n,\infty ))$ must be constant.     

Deformation of hypersurfaces preserving the Möbius metric and a reduction theorem

A hypersurface without umbilics in the (n+1)$(n+1)$-dimensional Euclidean space f:Mⁿ→Rⁿ⁺¹$f:M^n\to R^{n+1}$ is known to be determined by the Möbius metric g and the Möbius second fundamental form B   up to a Möbius transformation when n?3$n?3$. In this paper we consider Möbius rigidity for hypersurfaces and deformations of a hypersurface preserving the Möbius metric in the high dimensional case n?4$n?4$. When the highest multiplicity of principal curvatures is less than n−2$n-2$, the hypersurface is Möbius rigid. When the multiplicities of all principal curvatures are constant, deformable hypersurfaces and the possible deformations are also classified completely. In addition, we establish a reduction theorem characterizing the classical construction of cylinders, cones, and rotational hypersurfaces, which helps to find all the non-trivial deformable examples in our classification with wider application in the future.