A classification of \({\text {SL}}(n)\) and translation covariant Minkowski valuations on log-concave functions is established. The moment vector and the recently introduced level set body of log-concave functions are characterized. Furthermore, analogs of the Euler characteristic and volume are characterized as \({\text {SL}}(n)\) and translation invariant valuations on log-concave functions.

The method of alternating projections is used to examine how regularity of operators associated to the \({{\bar{\partial }}}\)-Neumann problem percolates up the \({{\bar{\partial }}}\)-complex. The approach revolves around operator identities—rather than estimates—that hold on any Lipschitz domain in \({{\mathbb {C}}}^n\), not necessarily bounded or pseudoconvex. We show that a geometric rate of convergence

We discuss the links between stationary discs, the defect of analytic discs, and 2-jet determination of CR automorphisms of generic nondegenerate real submanifolds of \({\mathbb C}^{N}\) of class \({\mathcal {C}}^{4}\).

It is shown that m disjoint sets with fixed Gaussian volumes that partition \(\mathbb {R}^{n}\) with minimum Gaussian surface area must be \((m-1)\)-dimensional. This follows from a second variation argument using infinitesimal translations. The special case \(m=3\) proves the Double Bubble problem for the Gaussian measure, with an extra technical assumption. That is, when \(m=3\), the three minimal

It has been known that sharp Sobolev embeddings into weak Lebesgue spaces are non-compact but the question of whether the measure of non-compactness of such an embedding equals to its operator norm constituted a well-known open problem. The existing theory suggested an argument that would possibly solve the problem should the target norms be disjointly superadditive, but the question of disjoint superadditivity

We minimize a linear combination of the length and the \(L^2\)-norm of the curvature among networks in \(\mathbb {R}^d\) belonging to a given class determined by the number of curves, the order of the junctions, and the angles between curves at the junctions. Since this class lacks compactness, we characterize the set of limits of sequences of networks bounded in energy, providing an explicit representation

In this paper, on Lie groups of polynomial growth, we make an estimation of the kernel function of multilinear spectral multipliers. Then as an application of this estimation, we prove the \(L^{p}\) boundedness and weighted \(L^p\) boundedness of such multilinear spectral multipliers.

We consider various definitions of functions of vanishing mean oscillation on a domain \(\Omega \subset {{{\mathbb {R}}}^n}\). If the domain is uniform, we show that there is a single extension operator which extends functions in these spaces to functions in the corresponding spaces on \({{{\mathbb {R}}}^n}\), and also extends \(\mathrm{BMO}(\Omega )\) to \(\mathrm{BMO}({{{\mathbb {R}}}^n})\), generalizing

Let M be a closed complex submanifold in \({\mathbb {C}}^N\) with the complete Kähler metric induced by the Euclidean metric. Several finiteness theorems on the \(L^p\) Bergman space of holomorphic sections of a given Hermitian line bundle L over M and the associated \(L^2\) cohomology groups are obtained. Some infiniteness theorems are also given in order to test the accuracy of finiteness theorems

We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual \(\ell ^1\)-sum in the sparse operator is replaced by an \(\ell ^r\)-sum. This sparse domination theorem is applicable to various

We study relations between special elliptic isometries in the complex hyperbolic plane. Relations of lengths 2, 3, and 4 are fully classified. Some relative \(\mathrm{SU}(2,1)\)-character varieties of the quadruply punctured sphere are described and applied to the study of length 5 relations.

In this paper, we consider some norm estimates for mixed Morrey spaces considered by the first author. Mixed Lebesgue spaces are realized as a special case of mixed Morrey spaces. What is new in this paper is a new norm estimate for mixed Morrey spaces that is applicable to mixed Lebesgue spaces as well. An example shows that the condition on parameters is optimal. As an application, the Olsen inequality

We define Cayley structures as a field of Cayley’s ruled cubic surfaces over a four dimensional manifold and motivate their study by showing their similarity to indefinite conformal structures and their link to differential equations and the theory of integrable systems. In particular, for Cayley structures an extension of certain notions defined for indefinite conformal structures in dimension four

We study the vector-valued spectrum \({\mathcal {M}}_\infty (B_{c_0},B_{c_0})\), that is, the set of nonzero algebra homomorphisms from \(\mathcal {H}^\infty (B_{c_0})\) to \(\mathcal {H}^\infty (B_{c_0})\) which is naturally projected onto the closed unit ball of \(\mathcal {H}^\infty (B_{c_0}, \ell _\infty )\), likewise the scalar-valued spectrum \(\mathcal {M}_\infty (B_{c_0})\) which is projected

Using a stability criterion due to Kröncke, we show, providing \({n\ne 2k}\), the Kähler–Einstein metric on the Grassmannian \(Gr_{k}(\mathbb {C}^{n})\) of complex k-planes in an n-dimensional complex vector space is dynamically unstable as a fixed point of the Ricci flow. This generalises the recent results of Kröncke and Knopf–Sesum on the instability of the Fubini–Study metric on \(\mathbb {CP}^{n}\)

We take the first step in the development of an equivariant version of modern, Gromov-style Oka theory. We define equivariant versions of the standard Oka property, ellipticity, and homotopy Runge property of complex manifolds, show that they satisfy all the expected basic properties, and present examples. Our main theorem is an equivariant Oka principle saying that if a finite group G acts on a Stein

We show that the set of Finsler metrics on a manifold contains an open everywhere dense subset of Finsler metrics with infinite-dimensional holonomy groups.

We consider a compact Riemann surface R of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate R into two subsets: a connected Riemann surface \(\Sigma \), and the union \(\mathcal {O}\) of a finite collection of simply connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on \(\mathcal {O}\) to the

We establish Thom’s jet transversality theorem for regular maps from an affine algebraic manifold to an algebraic manifold satisfying a suitable flexibility condition. It can be considered as the algebraic version of Forstnerič’s jet transversality theorem for holomorphic maps from a Stein manifold to an Oka manifold. Our jet transversality theorem implies genericity theorems for regular maps of maximal

We prove that the mappings obtained in Forstnerič splitting lemma vary in a \(\mathcal {C}^{\lfloor {\frac{l-1}{2}}\rfloor }\)-continuous way if only the input family of biholomorphic mappings close to Id (and their domains) is \(\mathcal {C}^l\)-continuous (see Theorem 1.3 for a precise formulation).

We give a moment map interpretation of some relatively balanced metrics. As an application, we extend a result of S. K. Donaldson on constant scalar curvature Kähler metrics to the case of extremal metrics. Namely, we show that a given extremal metric is the limit of some specific relatively balanced metrics. As a corollary, we recover uniqueness and splitting results for extremal metrics in the polarized

The sign uncertainty principle of Bourgain et al. asserts that if a function \(f:{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) and its Fourier transform \({\widehat{f}}\) are nonpositive at the origin and not identically zero, then they cannot both be nonnegative outside an arbitrarily small neighborhood of the origin. In this article, we establish some equivalent formulations of the sign uncertainty

We prove the weighted \(L^p\) regularity of the ordinary Bergman projection on certain pseudoconvex domains where the weight belongs to an appropriate generalization of the Békollè–Bonami class. The main tools used are estimates on the Bergman kernel obtained by McNeal and Békollè’s original approach of proving a good-lambda inequality.

We study several of the recent conjectures in regards to the role of symmetry in the inequalities of Brunn–Minkowski type, such as the \(L_p\)-Brunn–Minkowski conjecture of Böröczky, Lutwak, Yang and Zhang, and the Dimensional Brunn–Minkowski conjecture of Gardner and Zvavitch, in a unified framework. We obtain several new results for these conjectures. We show that when \(K\subset L,\) the multiplicative

In this article, we study \(C^1\) regular curves in the 2-sphere that start and end at given points with given directions, whose tangent vectors are Lipschitz continuous, and their a.e. existing geodesic curvatures have essentially bounds in an open interval. Especially, we show that a \(C^1\) regular curve is such a curve if and only if the infimum of its lower curvature and the supremum of its upper

It is well known that the Hilbert matrix \({\mathrm {H}}\) is bounded on weighted Bergman spaces \(A^p_\alpha \) if and only if \(1<\alpha +20\) and \(p\ge 2(\alpha +2)\), which reduces the conjecture in the case when \(\alpha >0\) to the interval \(\alpha +2\alpha +2\) there has been no progress so far in proving the conjecture, moreover, there is no even an explicit upper bound for the norm of the

In this paper, we study the minimizers of curvature functionals (Willmore functional and extrinsic energy functional) subject to an area constraint in asymptotically flat manifolds. Under some certain conditions, we prove that such minimizers exist. Besides the surface theory related to the Willmore functional, the proofs also rely on the inverse mean curvature flow developed by Huisken and Ilmanen

In this paper, we introduce the notion of r-trapped submanifolds immersed in generalized Robertson–Walker spacetimes as generalization of the trapped submanifolds introduced by Penrose. Considering some properties such as parabolicity and stochastic completeness, we prove rigidity and nonexistence results for r-trapped in some configurations of GRW spacetimes and, lastly, we provide examples of r-trapped

We consider the problem of the boundedness of maximal operators on \(\mathrm {BMO}_{}^{}\) on shapes in \({\mathbb {R}}^n\). We prove that for bases of shapes with an engulfing property, the corresponding maximal function is bounded from \(\mathrm {BMO}_{}^{}\) to \(\mathrm {BLO}_{}^{}\), generalising a known result of Bennett for the basis of cubes. When the basis of shapes does not possess an engulfing

We prove smoothness and provide the asymptotic behavior at infinity of solutions of Dirac–Einstein equations on \(\mathbb {R}^3\), which appear in the bubbling analysis of conformal Dirac–Einstein equations on spin 3-manifolds. Moreover, we classify ground state solutions, proving that the scalar part is given by Aubin–Talenti functions, while the spinorial part is the conformal image of \(-\frac{1}{2}\)-Killing

Let \(M^n\) be a complete n-dimensional Riemannian manifold and \(\Gamma _f\) the graph of a \(C^2\)-function f defined on a metric ball of \(M^n\). In the same spirit of the estimates obtained by Heinz for the mean and Gaussian curvatures of a surface in \({\mathbb {R}}^3\) which is a graph over an open disk in the plane, we obtain in this work upper estimates for \(\inf |R|\), \(\inf |A|\) and \(\inf

For a metric space X we study metrics on the two copies of X. We define composition of such metrics and show that the equivalence classes of metrics are a semigroup M(X). Our main result is that M(X) is an inverse semigroup. Therefore, one can define the \(C^*\)-algebra of this inverse semigroup, which is not necessarily commutative. If the Gromov–Hausdorff distance between two metric spaces, X and

We study the embeddings of variable exponent Sobolev and Hölder function spaces over Euclidean domains, providing necessary and/or sufficient conditions on the regularity of the exponent and/or the domain in various contexts. Concerning the exponent, the relevant condition is log-Hölder continuity; concerning the domain, the relevant condition is the measure density condition.

We develop almost-orthogonality principles for maximal functions associated with averages over line segments and directional singular integrals. Using them, we obtain sharp \(L^2\)-bounds for these maximal functions when the underlying direction set is equidistributed in \({\mathbb {S}}^{n-1}\).

A modified cylindrical Poisson–Schrödinger integral is constructed in terms of Nevanlinna norm associated with cylindrical Schrödinger equation, then with the help of Carleman–Schrödinger formula, explicit solutions of the equation mentioned above can be generated via the Schrödinger integral representation.

The Möbius energy is one of the knot energies, and is named after its Möbius invariant property. It is known to have several different expressions. One is in terms of the cosine of conformal angle, and is called the cosine formula. Another is the decomposition into Möbius invariant parts, called the decomposed Möbius energies. Hence the cosine formula is the sum of the decomposed energies. This raises

Inspired by the idea of Colding and Minicozzi (Ann Math 182:755–833, 2015), we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a closed ambient manifold with non-negative Ricci curvature. Moreover, this entropy is monotone along the mean curvature flow in a closed Riemannian

We show that a basis of a semisimple Lie algebra of compact type, for which any diagonal left-invariant metric has a diagonal Ricci tensor, is characterized by the Lie algebraic condition of being “nice”. Namely, the bracket of any two basis elements is a multiple of another basis element. This extends the work of Lauret and Will (Proc Am Math Soc 141(10):3651–3663, 2013) on nilpotent Lie algebras

Sparse representations of multidimensional data have received a significant attention in the literature due to their applications in problems of data restoration and feature extraction. In this paper, we consider an idealized class \({\mathcal {C}}^2(Z) \subset L^2({\mathbb {R}}^3)\) of 3-dimensional data dominated by surface singularities that are orthogonal to the xy plane. To deal with this type

Let \(m\ge 2, \lambda > 1\) and define the multilinear Littlewood–Paley–Stein operators by \( g_{\lambda ,\mu }^*(\vec {f})(x) = (\iint _{{\mathbb {R}}^{n+1}_{+}} (\frac{t}{t + |x - y|})^{m \lambda } |\int _{({{\mathbb {R}}^n})^{\kappa }} s_t(y,\vec {z}) \prod _{i=1}^{\kappa } f_i(z_i) \ \mathrm{d}\mu (z_1) \cdots \mathrm{d}\mu (z_{\kappa })|^2 \frac{\mathrm{d}\mu (y) \mathrm{d}t}{t^{m+1}})^{1/2}.\)

In this paper, we establish various maximal principles and develop the direct moving planes and sliding methods for equations involving the physically interesting (nonlocal) pseudo-relativistic Schrödinger operators \((-\Delta +m^{2})^{s}\) with \(s\in (0,1)\) and mass \(m>0\). As a consequence, we also derive multiple applications of these direct methods. For instance, we prove monotonicity, symmetry

Given a smooth polarized Riemann surface (X, L) endowed with a hyperbolic metric \(\omega \) that has standard cusp singularities along a divisor D, we show the \(L^2\) projective embedding of (X, D) defined by \(L^k\) is asymptotically almost balanced in a weighted sense. The proof depends on sufficiently precise understanding of the behavior of the Bergman kernel in three regions, with the most crucial

Following the work of Berman and Witt Nyström we present the proof of existence and uniqueness of solutions to transported Monge–Ampère problem on complex compact manifolds with completely integrable torus action. The proof is based on the real theory of optimal transportation and some convex analysis.

In this paper, we give estimates on partial derivatives and logarithmic partial derivatives for holomorphic functions on polydiscs. Estimates will also be utilized to characterize entire solutions of a class of partial differential equations, which gives a new form of Picard’s theorem and its higher dimensional version.

We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold \(M\times \mathbb {R}\), where M is asymptotically flat. If the initial hypersurface \(F_0\subset M\times \mathbb {R}\) is uniformly spacelike and asymptotic to \(M\times \left\{ s\right\} \) for some \(s\in \mathbb {R}\) at infinity, we show that a mean curvature flow starting

In this note we prove the strong unique continuation property at the origin for the solutions of the parabolic differential inequality $$\begin{aligned} |\Delta u - u_t| \le \frac{M}{|x|^2} |u|, \end{aligned}$$ with the critical inverse square potential. Our main result sharpens a previous one of Vessella concerned with the subcritical case.

This paper aims to define and study currents and slices of currents in the Heisenberg group \({\mathbb {H}}^n\). Currents, depending on their integration properties and on those of their boundaries, can be classified into subspaces and, assuming their support to be compact, we can work with currents of finite mass, define the notion of slices of Heisenberg currents and show some important properties

Suppose \(p\ge 1\), \(w=P[F]\) is a harmonic mapping of the unit disk \({\mathbb D}\) satisfying F is absolutely continuous and \({\dot{F}}\in L^p(0, 2\pi )\), where \({\dot{F}}(e^{it})=\frac{\mathrm {d}}{\mathrm {d}t}F(e^{it})\). In this paper, we obtain Bergman norm estimates of the partial derivatives for w, i.e., \(\Vert w_z\Vert _{L^p}\) and \(\Vert \overline{w_{{\bar{z}}}}\Vert _{L^p}\), where

The uniformization and hyperbolization transformations formulated by Bonk et al. in “Uniformizing Gromov Hyperbolic Spaces”, Astérisque, vol 270 (2001), dealt with geometric properties of metric spaces. In this paper we consider metric measure spaces and construct a parallel transformation of measures under the uniformization and hyperbolization procedures. We show that if a locally compact roughly

A mixed symmetric Chernoff type inequality for two convex domains can be proved, furthermore its stability properties can also be obtained in the Hausdorff and \(L_2\) metrics, respectively.

Let M be an open Riemann surface and let \(\varLambda \subset M\) be a closed discrete subset. In this paper, we prove the existence of complete conformal minimal immersions \(M\rightarrow {\mathbb {R}}^n\), \(n\ge 3\), with prescribed values on \(\varLambda \) and whose generalized Gauss map \(M\rightarrow \mathbb {CP}^{n-1}\), \(n\ge 3\), avoids n hyperplanes of \(\mathbb {CP}^{n-1}\) located in

The existence of nontrivial solutions for the following kind of Klein–Gordon–Maxwell system $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u-(2\omega +\phi )\phi u=f(x,u),&{}\quad x\in {{\mathbb {R}}}^{3},\\ \Delta \phi =(\omega +\phi )u^{2},&{}\quad x\in {{\mathbb {R}}}^{3}, \end{array}\right. \end{aligned}$$ is investigated, where \(\omega >0\) is a constant, \(V\in C({{\mathbb {R}}}^{3}

We construct non-trapping asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points

We study the regularity of minimizers of a two-phase free boundary problem. For a class of n-dimensional convex domains, we establish the Lipschitz continuity of the minimizer up to the fixed boundary under Neumann boundary conditions. Our proof uses an almost monotonicity formula for the Alt–Caffarelli–Friedman functional restricted to the convex domain. This requires a variant of the classical Friedland–Hayman

We characterise the real extreme points of the unit ball of \(m^0_\Psi \), the complex extreme points of the unit ball of \(m_\Psi \) and the real extreme and exposed points of the unit ball of \((m_\Psi ^0)'\). Using these characterisations we show that, depending on the length of the extreme points, the multipliers of \(m^0_\Psi \) are either constant multiple of the identity or diagonal operators

In a domain \(\Omega \subset {\mathbb {R}}^{n}\) whose boundary is smooth except on a set \({\mathcal {C}}\) of codimension (in \(\partial \Omega \)) k, the behavior of a nonparametric prescribed mean curvature hypersurface \(z=f\left( \mathbf{x}\right) \) in the vertical cylinder \(\Omega \times {\mathbb {R}}\) at a point \({\mathcal {O}}\in {\mathcal {C}}\) can be largely unknown when \(n\ge 3,\)

We give asymptotic estimates of the variance of the number of integer points in translated thin annuli in any dimension.

We classify all the translating solitons to the mean curvature flow in the three-dimensional Heisenberg group that are invariant under the action of some one-parameter group of isometries of the ambient manifold. We provide a complete classification for any canonical deformation of the standard Riemannian metric of the Heisenberg group. We highlight similarities and differences with the analogous Euclidean

We show that if an exponential polynomial \(\sum _{i=1}^m P_i(z)e^{Q_i(z)}\), where \(P_i\), \(Q_i\in \mathbb C[z]\), is a dth power, \(d\ge 2\), of an entire function g, then g itself is also an exponential polynomial. We also study when a multivariable polynomial with moving targets of slow growth evaluated at unit arguments can be a dth power of an entire function. Finally, we formulate a boundary