We prove the existence of complete cohomogeneity one triaxial Kähler–Einstein metrics in dimension four under an action of the Euclidean group E(2). We also demonstrate local existence of Ricci flat Kähler metrics of a related type that are given via generalized PDEs, and determine, under mild conditions, whether they are complete. The common framework for both metric types is a frame-dependent system

We prove that every J-holomorphic variational vector field can be realized as derivation \(\frac{d}{dt}_{|t=0}f_t\) where \((f_t)\) is a one parametric family of J-holomorphic discs. Furthermore, we discuss properness of an extremal J-holomorphic disc in a bounded pseudoconvex domain.

Embedded symmetry within the Heisenberg group is used to couple geometric insight and analytic calculation to obtain a new sharp Stein–Weiss inequality with mixed homogeneity on the line of duality. SL(2,R) invariance and Riesz potentials define a natural bridge for encoded information that connects distinct geometric structures. The intrinsic character of the Heisenberg group makes it the natural

A Radon measure \(\mu \) is n-rectifiable if it is absolutely continuous with respect to \({\mathcal {H}}^n\) and \(\mu \)-almost all of \({{\,\mathrm{supp}\,}}\mu \) can be covered by Lipschitz images of \({\mathbb {R}}^n\). In this paper we give two sufficient conditions for rectifiability, both in terms of square functions of flatness-quantifying coefficients. The first condition involves the so-called

The symmetrized bidisc $$\begin{aligned} G {\mathop {=}\limits ^\mathrm{{def}}}\{(z+w,zw):|z|<1,\quad |w|<1\}, \end{aligned}$$ under the Carathéodory metric, is a complex Finsler space of cohomogeneity 1 in which the geodesics, both real and complex, enjoy a rich geometry. As a Finsler manifold, G does not admit a natural notion of angle, but we nevertheless show that there is a notion of orthogonality

In this paper, we first obtain an \(L^q\) gradient estimate for p-harmonic maps, by assuming the target manifold supporting a certain function, whose gradient and Hessian satisfy some analysis conditions. From this \(L^q\) gradient estimate, we get a corresponding Liouville type result for p-harmonic maps. Secondly, using these general results, we give various geometric applications to p-harmonic maps

We prove the asymptotic of the logarithmic Bergman kernel. And as an application, we calculate the conditional expectation of density of zeros of Gaussian random sections of powers of a positive line bundle that vanish along a fixed smooth subvariety.

In this paper, we propose a new type of a neural network which is inspired by Gabor systems from harmonic analysis. In this regard, we construct a class of sparsely connected neural networks utilizing the concept of time–frequency shifts, and we show that their approximation error rates can be tied to the number of modulations in the corresponding Gabor frame and to the smoothness of the input function

We present a short historical overview of the Mikhlin–Hörmander and Marcinkiewicz multiplier theorems. We discuss different versions of them and provide comparisons. We also present a recent improvement of the Marcinkiewicz multiplier theorem in the two-dimensional case.

We show that the first twisted cohomology group associated with closed 1-forms on differentiable manifolds is related to certain 2-dimensional representations of the fundamental group. In particular, we construct examples of nowhere-vanishing 1-forms with non-trivial twisted cohomology.

Given any closed Riemannian manifold (M, g) we use the Lyapunov–Schmidt finite-dimensional reduction method and the classical Morse and Lusternick–Schnirelmann theories to prove multiplicity results for positive solutions of a subcritical Yamabe type equation on (M, g). If (N, h) is a closed Riemannian manifold of constant positive scalar curvature we obtain multiplicity results for the Yamabe equation

The “noncommutative graphs” which arise in quantum error correction are a special case of the quantum relations introduced in Weaver (Quantum relations. Mem Am Math Soc 215(v–vi):81–140, 2012). We use this perspective to interpret the Knill–Laflamme error-correction conditions (Knill and Laflamme in Theory of quantum error-correcting codes. Phys Rev A 55:900-911, 1997) in terms of graph-theoretic independence

In this article we extend to arbitrary p-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case \(p=2\). We first show that the set of singular points of such a map can be quantitatively stratified: we classify singular points based on the number of almost-symmetries of the map around them, as done in Cheeger and Naber (Commun Pure Appl Math 66(6):

We assume that \(n\ge 3\), \(u \in C^{2}( \mathbb {B}^{n},\mathbb {R}^{n}) \cap C(\overline{\mathbb {B}}^{n},\mathbb {R}^{n})\) is a solution to the hyperbolic Poisson equation \(\Delta _{h}u=\psi \) in \(\mathbb {B}^{n}\) with the boundary condition \(u|_{\mathbb {S}^{n-1}}=\phi \), where \(\Delta _{h}\) is the hyperbolic Laplace operator and \(\psi \in C( \mathbb {B}^{n},\mathbb {R}^{n})\). In Chen

If \(\mu \) is an arbitrary bounded Radon measure \(\mu \) on \({\mathbb {R}}^n,\) we denote by \({{\widehat{\mu }}}\) the Fourier–Stieltjes transform of \(\mu \) and by \(\sigma \) the pure point part of \(\mu .\) A closed \(\varLambda \subset {{\mathbb {R}}}^n\) is a gregarious set if the following property is satisfied: $$\begin{aligned} (\forall \mu )\, {{\widehat{\mu }}}=0\quad \mathrm{on}\, \varLambda

First we introduce the notions of \(\eta \)-parallel and \(\eta \)-commuting shape operator for real hypersurfaces in the complex quadric \(Q^m = SO_{m+2}/SO_mSO_2\). Next we give a complete classification of real hypersurfaces in the complex quadric \(Q^m\) with such kind of shape operators. By virtue of this classification we give a new characterization of ruled real hypersurface foliated by complex

In this paper, we consider some rigidity results for the Einstein metrics as the critical points of some known quadratic curvature functionals on complete manifolds, characterized by some point-wise inequalities. Moreover, we also provide rigidity results by the integral inequalities involving the Weyl curvature, the traceless Ricci curvature and the Sobolev constant, accordingly.

We classify complete curvature homogeneous metrics on simply connected four-dimensional manifolds which are invariant under a cohomogeneity one action.

Applying the equivariant moving frames method, we construct a convergent normal form for real-analytic 5-dimensional totally nondegenerate submanifolds of \({\mathbb {C}}^4\). We develop this construction by applying further normalizations, the possibility of which completely relies upon vanishing/non-vanishing of some specific coefficients of the normal form. This in turn divides the class of our

We study the Littlewood–Paley–Stein functions associated with Hodge-de Rham and Schrödinger operators on Riemannian manifolds. Under conditions on the Ricci curvature, we prove their boundedness on \(L^p\) for p in some interval \((p_1,2]\) and make a link to the Riesz Transform. An important fact is that we do not make assumptions of doubling measure or estimates on the heat kernel in this case. For

We study upper bounds for the first non-zero eigenvalue of the Steklov problem defined on finite graphs with boundary. For finite graphs with boundary included in a Cayley graph associated to a group of polynomial growth, we give an upper bound for the first non-zero Steklov eigenvalue depending on the number of vertices of the graph and of its boundary. As a corollary, if the graph with boundary also

In the present paper, we show how to define suitable subgroups of the orthogonal group \({O}(d-m)\) related to the unbounded part of a strip-like domain \(\omega \times {\mathbb {R}}^{d-m}\) with \(d\ge m+2\), in order to get “mutually disjoint” nontrivial subspaces of partially symmetric functions of \(H^1_0(\omega \times {\mathbb {R}}^{d-m})\) which are compactly embedded in the associated Lebesgue

The purpose of this paper is to investigate the geometric properties of real hypersurfaces of D’Angelo infinite type in \({{\mathbb {C}}}^n\). In order to understand the situation of flatness of these hypersurfaces, it is natural to ask whether there exists a nonconstant holomorphic curve tangent to a given hypersurface to infinite order. A sufficient condition for this existence is given by using

The original version of the article unfortunately contained an error in the acknowledgments section. The corrected Acknowledgements is given below.

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.