Journal of Geometric Analysis最新文献

We prove that constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class are K-semistable, naturally generalising the situation for polarised manifolds. Relying on a recent result by R. Berman, T. Darvas and C. Lu regarding properness of the K-energy, it moreover follows that cscK manifolds with discrete automorphism group are uniformly K-stable. As a main step of the proof we establish, in the general Kähler setting, a formula relating the (generalised) Donaldson-Futaki invariant to the asymptotic slope of the K-energy along weak geodesic rays.

Given a projective structure on a surface $N$ , we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature as well as a symplectic form on the total space M of a certain rank 2 affine bundle $M\to N$ . The Einstein metric has anti-self-dual conformal curvature and admits a parallel field of anti-self-dual planes. We show that locally every such metric arises from our construction unless it is conformally flat. The homogeneous Einstein metric corresponding to the flat projective structure on ${\mathrm{RP}}^2$ is the non-compact real form of the Fubini-Study metric on $M=\mathrm{SL}(3,R)/\mathrm{GL}(2,R)$ . We also show how our construction relates to a certain gauge-theoretic equation introduced by Calderbank.

We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw-Sommerfeld's heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.

We propose a framework for bilinear multiplier operators defined via the (bivariate) spectral theorem. Under this framework, we prove Coifman-Meyer type multiplier theorems and fractional Leibniz rules. Our theory applies to bilinear multipliers associated with the discrete Laplacian on $Z^d,$ general bi-radial bilinear Dunkl multipliers, and to bilinear multipliers associated with the Jacobi expansions.

We prove that given a family $\left(G_t\right)$ of strictly pseudoconvex domains varying in $C^2$ topology on domains, there exists a continuously varying family of peak functions $h_{t,\zeta }$ for all $G_t$ at every $\zeta \in \partial G_t$ .

We investigate the existence of wandering Fatou components for polynomial skew-products in two complex variables. In 2004, the non-existence of wandering domains near a super-attracting invariant fiber was shown in Lilov (Fatou theory in two dimensions, PhD thesis, University of Michigan, 2004). In 2014, it was shown in Astorg et al. (Ann Math, arXiv:1411.1188 [math.DS], 2014) that wandering domains can exist near a parabolic invariant fiber. In Peters and Vivas (Math Z, arXiv:1408.0498, 2014), the geometrically attracting case was studied, and we continue this study here. We prove the non-existence of wandering domains for subhyperbolic attracting skew-products; this class contains the maps studied in Peters and Vivas (Math Z, arXiv:1408.0498, 2014). Using expansion properties on the Julia set in the invariant fiber, we prove bounds on the rate of escape of critical orbits in almost all fibers. Our main tool in describing these critical orbits is a possibly singular linearization map of unstable manifolds.

For the natural two-parameter filtration $\left(F_{\lambda },:,\lambda \in P\right)$ on the boundary of a triangle building, we define a maximal function and a square function and show their boundedness on $L^p\left({\Omega }_0\right)$ for $p\in (1,\infty )$ . At the end, we consider $L^p\left({\Omega }_0\right)$ boundedness of martingale transforms. If the building is of $\text{GL}(3,Q_p)$ , then ${\Omega }_0$ can be identified with p-adic Heisenberg group.

Let $M(n,D)$ be the space of closed n-dimensional Riemannian manifolds (M, g) with $\mathrm{diam}\left(M\right)\le D$ and $|{sec}^M|\le 1$ . In this paper we consider sequences $(M_i,g_i)$ in $M(n,D)$ converging in the Gromov-Hausdorff topology to a compact metric space Y. We show, on the one hand, that the limit space of this sequence has at most codimension one if there is a positive number r such that the quotient $\frac{\mathrm{vol}\left(B_r^{M_i}\left(x\right)\right)}{{\mathrm{inj}}^{M_i}\left(x\right)}$ can be uniformly bounded from below by a positive constant C(n, r, Y) for all points $x\in M_i$ . On the other hand, we show that if the limit space has at most codimension one then for all positive r there is a positive constant C(n, r, Y) bounding the quotient $\frac{\mathrm{vol}\left(B_r^{M_i}\left(x\right)\right)}{{\mathrm{inj}}^{M_i}\left(x\right)}$ uniformly from below for all $x\in M_i$ . As a conclusion, we derive a uniform lower bound on the volume and a bound on the essential supremum of the sectional curvature for the closure of the space consisting of all manifolds in $M(n,D)$ with $C\le \frac{\mathrm{vol}\left(M\right)}{\mathrm{inj}\left(M\right)}$ .

We prove Luzin N- and Morse-Sard properties for mappings $v:R^n\to R^d$ of the Sobolev-Lorentz class $W_{p,1}^k$ , $p=\frac{n}{k}$ (this is the sharp case that guaranties the continuity of mappings). Our main tool is a new trace theorem for Riesz potentials of Lorentz functions for the limiting case $q=p$ . Using these results, we find also some very natural approximation and differentiability properties for functions in $W_{p,1}^k$ with exceptional set of small Hausdorff content.