We prove that a 4-dimensional \(C^2\) conformally compact Einstein manifold with Hölder continuous scalar curvature and with \(C^{m,\alpha }\) boundary metric has a \(C^{m,\alpha }\) compactification. We also study the regularity of the new structure and the new defining function. This is a supplementary proof of Anderson’s work and an improvement of Helliwell’s result in dimension 4.

In this paper, we introduce a horizontal F-energy functional for maps from a Riemannian manifold to a Carnot group. The critical maps of this functional are referred to as CC-F-harmonic maps. Under suitable conditions on the Hessian of the distance function and the degree of F(t), we obtain several Liouville-type theorems for CC-F-harmonic maps from some complete Riemannian manifolds to Carnot groups

In 2006 Carbery raised a question about an improvement on the naïve norm inequality \(\Vert f+g\Vert _p^p \le 2^{p-1}(\Vert f\Vert _p^p + \Vert g\Vert _p^p)\) for two functions f and g in \(L^p\) of any measure space. When \(f=g\) this is an equality, but when the supports of f and g are disjoint the factor \(2^{p-1}\) is not needed. Carbery’s question concerns a proposed interpolation between the

In this paper we study spectral stability of the \(\bar{\partial }\)-Neumann Laplacian under the Kohn–Nirenberg elliptic regularization. We obtain quantitative estimates for stability of the spectrum of the \(\bar{\partial }\)-Neumann Laplacian when either the operator or the underlying domain is perturbed.

We prove that any complete immersed globally orientable uniformly 2-convex translating soliton \(\Sigma \subset {\mathbb {R}}^{n+1}\) for the mean curvature flow is locally strictly convex. It follows that a uniformly 2-convex entire graphical translating soliton in \({\mathbb {R}}^{n+1},\, n\ge 3 \) is the axisymmetric “bowl soliton.”

We propose to study positive harmonic functions satisfying a nonlinear Neuman condition on a compact Riemannian manifold with nonnegative Ricci curvature and strictly convex boundary. A precise conjecture is formulated. We discuss its implications and present some partial results. Related questions are discussed for compact Riemannian manifolds with positive Ricci curvature and convex boundary.

We study the density of polynomials in \(H^2(E,\varphi )\), the space of square integrable functions with respect to \(\mathrm{e}^{-\varphi }\mathrm{d}m\) and holomorphic on the interior of E in \({\mathbb {C}}\), where \(\varphi \) is a subharmonic function and dm is a measure on E. We give a result where E is the union of a Lipschitz graph and a Carathéodory domain, which we state as a weighted \(L^2\)-version

The Hilbert metric on convex subsets of \({\mathbb {R}}^n\) has proven a rich notion and has been extensively studied. We propose here a generalization of this metric to subsets of complex projective spaces (see also L. Dubois in J Lond Math Soc (2) 79(3):719–737, 2009) and give examples of geometric applications. Basic examples include the hyperbolic metric on complex hyperbolic spaces, the n-punctured

In this short paper, we derive an alternative proof for some known (van den Berg & Gilkey 2015) short-time asymptotics of the heat content in a compact full-dimensional submanifolds S with smooth boundary. This includes formulae like $$\begin{aligned} \int _{S} \exp (t\Delta ) (f \mathbb {1}_{S}) \,\mathrm {d}V= \int _S f \,\mathrm {d}V- \sqrt{\frac{t}{\pi }} \int _{\partial S} f \,\mathrm {d}A+ o(\sqrt{t})

For \(\alpha \in {\mathbb {R}}\), let \(W_{\alpha }\) (respectively, \(w_{\alpha }\)) be the corresponding power weight function on \({\mathbb {R}}\) (respectively, the corresponding periodic power weight function, also defined on \({\mathbb {R}} \)). It is known that \(W_{\alpha }\) (respectively, \(w_{\alpha }\)) belongs to \(A_{1}\left( {\mathbb {R}}\right) \) (respectively, \(A_{1}\left( {\mathbb

In this paper, we consider the martingale Hardy spaces defined with the help of the mixed \(L_{\overrightarrow{p}}\)-norm. Five mixed martingale Hardy spaces will be investigated: \(H_{\overrightarrow{p}}^{s}\), \(H_{\overrightarrow{p}}^S\), \(H_{\overrightarrow{p}}^M\), \(\mathcal {P}_{\overrightarrow{p}}\), and \(\mathcal {Q}_{\overrightarrow{p}}\). Several results are proved for these spaces, like

Our primary objective in this paper is to study a class of k-Hessian equations with zero-order and first-order terms in unbounded domains. Among others, we derive Phragmén–Lindelöf and Liouville type results. The zero-order terms involved are required to satisfy Keller–Osserman type conditions.

We develop a new semiclassical calculus in analytic regularity, and apply these techniques to the study of Berezin–Toeplitz quantization in real-analytic regularity. We provide asymptotic formulas for the Bergman projector and Berezin–Toeplitz operators on a compact Kähler manifold. These objects depend on an integer N and we study, in the limit \(N\rightarrow +\infty \), situations in which one can

Abstract We prove local Lipschitz regularity and non-degeneracy estimates for viscosity solutions to one-phase free boundary problems governed by non-homogeneous fully non-linear equations with unbounded ingredients and quadratic growth in the gradient. We use the approach in De Silva (Interfaces Free Bound 13:223–238, 2011) to show that flat or Lipschitz free boundaries are graphs \(C^{1, \alpha }\)

Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar function on a globally hyperbolic stationary Lorentzian manifold and prove injectivity

In this paper, we establish some estimates of level sets of holomorphic functions. Relying on obtained estimates we compute some of the weighted log canonical thresholds of plurisubharmonic functions. Finally, we prove the analyticity of the sublevel sets of weighted log canonical thresholds of plurisubharmonic functions.

If X is a projective variety and G is an algebraic group acting algebraically on X, we provide a counter-example to the existence of a G-equivariant extension on the formal semi-universal deformation of X.

In this paper, we prove a classification for complete embedded constant weighted mean curvature hypersurfaces \(\varSigma \subset {\mathbb {R}}^{n+1}. \) We characterize the hyperplanes and generalized round cylinders by using an intrinsic property on the norm of the second fundamental form. Furthermore, we prove an equivalence of properness, finite weighted volume, and exponential volume growth for

This paper may be viewed as a companion paper to Greenblatt (arXiv:1910.04547). In that paper, \(L^2\) Sobolev estimates derived from a Newton polyhedron-based resolution of singularities method are combined with interpolation arguments to prove \(L^p\) to \(L^q_s\) estimates, some sharp up to endpoints, for translation invariant Radon transforms over hypersurfaces and related operators. Here \(q \ge

This paper is mainly concerned with the existence of extremals for the fractional Caffarelli–Kohn–Nirenberg and Trudinger–Moser inequalities. We first establish the existence of extremals of the fractional Caffarelli–Kohn–Nirenberg inequalities in Theorem 1.1. It is also proved that the extremals of this inequality are the ground-state solutions of some fractional p-Laplacian equation. Then, we use

In this paper, we prove local existence of a Ricci de Turck flow starting at a space with incomplete edge singularities and flowing for a short time within a class of incomplete edge manifolds. We derive regularity properties for the corresponding family of Riemannian metrics and discuss boundedness of the Ricci curvature along the flow. For Riemannian metrics that are sufficiently close to a flat

A celebrated theorem of Kanai states that quasi-isometries preserve isoperimetric inequalities between uniform Riemannian manifolds (with positive injectivity radius) and graphs. Our main result states that we can study the (Cheeger) isoperimetric inequality in a Riemann surface by using a graph related to it, even if the surface has injectivity radius zero (this graph is inspired in Kanai’s graph

We study the index of the APS boundary value problem for a strongly Callias-type operator \({\mathcal {D}}\) on a complete Riemannian manifold M. We show that this index is equal to an index on a simpler manifold whose boundary is a disjoint union of two complete manifolds \(N_0\) and \(N_1\). If the dimension of M is odd we show that the latter index depends only on the restrictions \({\mathcal {A}}_0\)

A parabolic cylinder is an invariant, non-recurrent Fatou component \(\Omega \) of an automorphism F of \(\mathbb {C}^2\) satisfying: (1) The closure of the \(\omega \)-limit set of F on \(\Omega \) contains an isolated fixed point, (2) there exists a univalent map \(\Phi \) from \(\Omega \) into \({\mathbb {C}}^2\) conjugating F to the translation \((z,w) \mapsto (z+1, w)\), and (3) every limit map

The center of mass of a finite measure with respect to a radially increasing weight is shown to exist, be unique, and depend continuously on the measure.

In this paper, we consider critical maps of a horizontal energy functional for maps from a sub-Riemannian manifold to a Riemannian manifold. These critical maps are referred to as subelliptic harmonic maps. In terms of the subelliptic harmonic map heat flow, we investigate the existence problem for subelliptic harmonic maps. Under the assumption that the target Riemannian manifold has non-positive

We study the area preserving Willmore flow in an asymptotic region of an asymptotically flat manifold which is \(C^3\)-close to Schwarzschild. It was shown by Lamm, Metzger and Schulze that such a region is foliated by spheres of Willmore type, see (Lamm et al. in Math Ann 350(1):1–78, 2011). In this paper, we prove that the leaves of this foliation are stable under small area preserving \(W^{2,2}\)-perturbations

We study an irregular double obstacle problem with Orlicz growth over a nonsmooth bounded domain. We establish a global Calderón–Zygmund estimate by proving that the gradient of the solution to such a nonlinear elliptic problem is as integrable as both the nonhomogeneous term in divergence form and the gradient of the associated double obstacles. We also investigate minimal regularity requirements

Let denote a finite circular sector of opening angle and radius one, and let denote the heat operator associated to the Dirichlet extension of the Laplacian

This paper is devoted to the study of the medial axes of sets definable in polynomially bounded o-minimal structures, i.e. the sets of points with more than one closest point with respect to the Euclidean distance. Our point of view is that of singularity theory. While trying to make the paper self-contained, we gather here also a large bunch of basic results. Our main interest, however, goes to the

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

We prove that given a family ( G t ) of strictly pseudoconvex domains varying in C 2 topology on domains, there exists a continuously varying family of peak functions h t , ζ for all G t at every ζ ∈ ∂ G t .

For the natural two-parameter filtration F λ : λ ∈ P on the boundary of a triangle building, we define a maximal function and a square function and show their boundedness on L p ( Ω 0 ) for p ∈ ( 1 , ∞ ) . At the end, we consider L p ( Ω 0 ) boundedness of martingale transforms. If the building is of GL ( 3 , Q p ) , then Ω 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 diam ( M ) ≤ D and | sec M | ≤ 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

We study the asymptotic behaviour of the partial density function associated to sections of a positive hermitian line bundle that vanish to a particular order along a fixed divisor Y. Assuming the data in question is invariant under an S 1 -action (locally around Y) we prove that this density function has a distributional asymptotic expansion that is in fact smooth upon passing to a suitable real blow-up

We prove that 2 d , 2 d - 3 ( d - 1 ) 2 , 2 d - 1 d ( d - 1 ) , 2 d - 5 d 2 - 3 d + 1 and 2 d - 3 d ( d - 2 ) are the smallest log canonical thresholds of reduced plane curves of degree d ⩾ 3 , and we describe reduced plane curves of degree d whose log canonical thresholds are these numbers. As an application, we prove that 2 d , 2 d - 3 ( d - 1 ) 2 , 2 d - 1 d ( d - 1 ) , 2 d - 5 d 2 - 3 d + 1 and

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature bounded below and diameter bounded above, Gromov-Hausdorff convergence agrees with intrinsic flat convergence. In particular, the limiting current is essentially unique, has multiplicity one, and mass equal to the Hausdorff measure. Moreover, the limit spaces satisfy a constancy theorem

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 → 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

We investigate problems related with the existence of square integrable holomorphic functions on (unbounded) balanced domains. In particular, we solve the problem of Wiegerinck for balanced domains in dimension two. We also give a description of L h 2 -domains of holomorphy in the class of balanced domains and present a purely algebraic criterion for homogeneous polynomials to be square integrable

We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isometric immersions of the Riemannian manifolds, independent of the local coordinates

We prove Luzin N- and Morse-Sard properties for mappings v : R n → R d of the Sobolev-Lorentz class W p , 1 k , p = 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

We deal with Morrey spaces on bounded domains Ω obtained by different approaches. In particular, we consider three settings M u , p ( Ω ) , M u , p ( Ω ) and M u , p ( Ω ) , where 0 < p ≤ u < ∞ , commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes E G ( M u , p ( Ω ) ) as well as E G ( M u , p ( Ω ) ) , and obtain some applications

We consider operators T satisfying a sparse domination property | ⟨ T f , g ⟩ | ≤ c ∑ Q ∈ S ⟨ f ⟩ p 0 , Q ⟨ g ⟩ q 0 ' , Q | Q | with averaging exponents 1 ≤ p 0 < q 0 ≤ ∞ . We prove weighted strong type boundedness for p 0 < p < q 0 and use new techniques to prove weighted weak type ( p 0 , p 0 ) boundedness with quantitative mixed A 1 - A ∞ estimates, generalizing results of Lerner, Ombrosi, and Pérez

In this paper, we get an inequality in terms of holomorphic sectional curvature of complex Finsler metrics. As applications, we prove a Schwarz Lemma from a complete Riemannian manifold to a complex Finsler manifold. We also show that a strongly pseudoconvex complex Finsler manifold with semi-positive but not identically zero holomorphic sectional curvature has negative Kodaira dimension under an extra

Let (M, g) be an analytic, compact, Riemannian manifold with boundary, of dimension n ≥ 2 . We study a class of generalized Radon transforms, integrating over a family of hypersurfaces embedded in M, satisfying the Bolker condition (in: Quinto, Proceedings of conference "Seventy-five Years of Radon Transforms", Hong Kong, 1994). Using analytic microlocal analysis, we prove a microlocal regularity theorem

We study the geometry of m-regular domains within the Caffarelli-Nirenberg-Spruck model in terms of barrier functions, envelopes, exhaustion functions, and Jensen measures. We prove among other things that every m-hyperconvex domain admits an exhaustion function that is negative, smooth, strictly m-subharmonic, and has bounded m-Hessian measure.

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 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 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

We construct positive singular solutions for the problem \(-\Delta u=\lambda \exp (e^u)\) in \(B_1\subset {\mathbb {R}}^n\) (\(n\ge 3\)), \(u=0\) on \(\partial B_1\), having a prescribed behaviour around the origin. Our study extends the one in Miyamoto (J Differ Equ 264:2684–2707, 2018) for such nonlinearities. Our approach is then carried out to elliptic equations featuring iterated exponentials

Optimal second-order regularity in the space variables is established for solutions to Cauchy–Dirichlet problems for nonlinear parabolic equations and systems of p-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the

In Saji et al. (J Math 62:259–280, 2008; Ann Math 169:491–529, 2009; J Geom Anal 222):383–409, 2012) the Gauss–Bonnet formulas for coherent tangent bundles over compact-oriented surfaces (without boundary) were proved. We establish the Gauss–Bonnet theorem for coherent tangent bundles over compact-oriented surfaces with boundary. We apply this theorem to investigate global properties of maps between

In this paper, we give an (integrated) \(p(>1)\)-Bochner–Weitzenböck formula and the p-Reilly type formula on Finsler manifolds. As applications, we obtain the p-Poincaré inequality on an n-dimensional compact Finsler manifold without boundary or with convex boundary under the assumption that Ricci\(_N\ge K\) for \(N\in [n, \infty ]\) and \(K\in \mathbb {R}\). In fact, we have the sharp p-Poincaré

It is remarkable that there is a duality in geometric tomography between results on projections of convex bodies and results on sections of star (rather than convex) bodies. The radial Blaschke addition, which is the dual version of Blaschke addition, as an operation between central symmetric star bodies is introduced in this paper. The relationship between it and the classical radial addition, many

We consider a complete non-compact Riemannian manifold satisfying the volume doubling property and a Gaussian upper bound for its heat kernel (on functions). Let \(\overrightarrow{\Delta }_k \) be the Hodge–de Rham Laplacian on differential k-forms with \(k \ge 1\). By the Bochner decomposition formula, \(\overrightarrow{\Delta }_k = \nabla ^* \nabla + R_k\), where \(\nabla \) denotes the Levi-Civita

This paper deals with a \(1/\kappa \)-type nonlocal flow for an initial convex closed curve \(\gamma _{0}\subset {\mathbb {R}}^{2}\) which preserves the convexity and the integral\(\ \int _{X\left( \cdot ,t\right) }\kappa ^{\alpha +1}ds,\ \alpha \in \left( -\infty ,\infty \right) ,\) of the evolving curve \(X\left( \cdot ,t\right) \). For\(\ \alpha \in [1,\infty ),\ \)it is proved that this flow exists

Motivated by questions related to the compactness of the \({\overline{\partial }}\)-Neumann operator, we study the restriction operator from the Bergman space of a domain in \(\mathbb {C}^n\) to the Bergman space of a non-empty open subset of the domain. We relate the restriction operator to the Toeplitz operator on the Bergman space of the domain whose symbol is the characteristic function of the

Guralnick (Linear Algebr Appl 99:85–96, 1988) proved that two holomorphic matrices on a noncompact connected Riemann surface, which are locally holomorphically similar, are globally holomorphically similar. We generalize this to (possibly, nonsmooth) one-dimensional Stein spaces. For Stein spaces of arbitrary dimension, we prove that global \(\mathcal {C}^\infty \) similarity implies global holomorphic

Complex Legendre duality is a generalization of Legendre transformation from Euclidean spaces to Kähler manifolds that Berndtsson, Cordero-Erausquin, Klartag, and Rubinstein have recently constructed. It is a local isometry of the space of Kähler potentials. We show that the fixed point of such a transformation must correspond to a real analytic Kähler metric.

A Riemannian metric on a compact 4-manifold is said to be Bach-flat if it is a critical point for the \(L^2\)-norm of the Weyl curvature. When the Riemannian 4-manifold in question is a Kähler surface, we provide a rough classification of solutions, followed by detailed results regarding each case in the classification. The most mysterious case prominently involves 3-dimensional CR manifolds.