We construct singular solutions to the Yamabe equation using a reduction of the problem in an equivariant setting. This provides a non-trivial geometric example for which the analysis is simpler than in Mazzeo-Pacard program. Our construction provides also a non-trivial example of a weak solution to the Yamabe problem involving an equation with (smooth) coefficients.

Given discrete groups \(\Gamma \subset \Delta \) we characterize \((\Gamma ,\sigma )\)-invariant spaces that are also invariant under \(\Delta \). This will be done in terms of subspaces that we define using an appropriate Zak transform and a particular partition of the underlying group. On the way, we obtain a new characterization of principal \((\Gamma ,\sigma )\)-invariant spaces in terms of the

We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on \(\textsf {CAT} (\kappa )\)-spaces and prove that they can be characterized by the same differential inclusion \(y_t'\in -\partial ^-\textsf {E} (y_t)\) one uses in the smooth setting and more precisely that \(y_t'\) selects the element of minimal norm in \(-\partial ^-\textsf {E} (y_t)\). This

This paper is concerned with the behavior of twisted Ruelle zeta functions of compact hyperbolic manifolds at the origin. Fried proved that for an orthogonal acyclic representation of the fundamental group of a compact hyperbolic manifold, the twisted Ruelle zeta function is holomorphic at \(s=0\) and its value at \(s=0\) equals the Reidemeister torsion. He also established a more general result for

We show that the set of cluster points of jumping numbers of a toric plurisubharmonic function in \(\mathbf {C}^n\) is discrete for every \(n \ge 1\). We also give a precise characterization of the set of those cluster points. These generalize a recent result of D. Kim and H. Seo from \(n=2\) to arbitrary dimension. Our method is to analyze the asymptotic behaviors of Newton convex bodies associated

Let \(X\subset {\mathbb {R}}^n\) be a (global) real analytic surface. Then every positive semidefinite meromorphic function on X is a sum of 10 squares of meromorphic functions on X. As a consequence, we provide a real Nullstellensatz for (global) real analytic surfaces.

We show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.

We characterize twisted convolutions associated with the Pedersen transform for unitary irreducible representations of nilpotent Lie groups. For \(1\le p<\infty \), we also prove the \(L^p\)-boundedness for the Pedersen \(L^p\)-multipliers in the case of unitary irreducible representations that are square-integrable modulo the center of the group under consideration, thus, generalizing an earlier result

We study the Hartogs extension phenomenon in non-compact toric varieties and its relation to the first cohomology group with compact support. We show that a toric variety admits this phenomenon if at least one connected component of the fan complement is concave, proving by this an earlier conjecture M. Marciniak.

We prove that, for a Finsler space, if the weighted Ricci curvature is bounded below by a positive number and the diam attains its maximal value, then it is isometric to a standard Finsler sphere. As an application, we show that the first eigenvalue of the Finsler-Laplacian attains its lower bound if and only if the Finsler manifold is isometric to a standard Finsler sphere, and moreover, we obtain

We prove several results on homogeneous plurisubharmonic polynomials on \({\mathbb {C}}^n\), \(n\in {\mathbb {Z}}_{\ge 2}\). Said results are relevant to the problem of constructing local bumpings at boundary points of pseudoconvex domains of finite D’Angelo 1-type in \({\mathbb {C}}^{n+1}\).

Let \({\mathbb {H}}^{n}={\mathbb {C}}^{n}\times {\mathbb {R}}\) be the n-dimensional Heisenberg group, \(Q=2n+2\) be the homogeneous dimension of \({\mathbb {H}}^{n}\). We establish in this paper that the following sharpened Trudinger–Moser inequalities on the Heisenberg group \({\mathbb {H}}^{n} \) under the homogeneous constraints of the Sobolev norm: $$\begin{aligned} \sup _{\Vert \nabla _{{\mathbb

We study the possible Hausdorff limits of the Julia sets and filled Julia sets of subsequences of the sequence of dual Chebyshev polynomials of a non-polar compact set \(K\subset {\mathbb C}\) and compare such limits to K. Moreover, we prove that the measures of maximal entropy for the sequence of dual Chebyshev polynomials of K converges weak* to the equilibrium measure on K.

In this paper, we study the existence and asymptotic behavior of localized nodal solutions for the following Kirchhoff-type equation $$\begin{aligned} -\left( \varepsilon ^2a+\varepsilon b\int _{\mathbb {R}^3}|\nabla u|^2dx\right) \Delta u+V(x)u=|u|^{p-2}u, \ x \in \mathbb {R}^3, \end{aligned}$$ where \(a>0\), \(b>0\) and \(40\) is sufficiently small, we construct the existence of a sequence of localized

A remarkable theorem of Joris states that a function f is \(C^\infty \) if two relatively prime powers of f are \(C^\infty \). Recently, Thilliez showed that an analogous theorem holds in Denjoy–Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris’s result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for

We establish a connection between the function space \(\mathrm{BMO}\) and the theory of quasisymmetric mappings on spaces of homogeneous type \({\widetilde{X}} :=(X,\rho ,\mu )\). The connection is that the logarithm of the generalised Jacobian of an \(\eta \)-quasisymmetric mapping \(f: {\widetilde{X}} \rightarrow {\widetilde{X}}\) is always in \(\mathrm{BMO}({\widetilde{X}})\). In the course of proving

We study extremal properties of the determinant of Friedrichs selfadjoint Laplacian on the Euclidean isosceles triangle envelopes of fixed area as a function of angles. We deduce an explicit closed formula for the determinant. Small-angle asymptotics show that the determinant grows without any bound as an angle of a triangle envelope goes to zero. We prove that the equilateral triangle envelope (the

This paper concerns the following magnetic Schrödinger–Poisson system $$\begin{aligned} {\left\{ \begin{array}{ll} -(\nabla +i A(x))^{2}u+(\lambda V(x)+Z(x))u+\phi u=f(\left| u\right| ^{2})u ,&{} \text { in }{{\mathbb {R}}}^{3}, \\ -\varDelta \phi = u^{2}, &{} \text { in } {\mathbb {R}}^{3}, \end{array}\right. } \end{aligned}$$ where \(\lambda \) is a positive parameter, f has subcritical growth, the

Slice Fueter-regular functions, originally called slice Dirac-regular functions, are generalized holomorphic functions defined over the octonion algebra \({\mathbb {O}}\), recently introduced by M. Jin, G. Ren and I. Sabadini. A function \(f:\Omega _D\subset {\mathbb {O}}\rightarrow {\mathbb {O}}\) is called (quaternionic) slice Fueter-regular if, given any quaternionic subalgebra \({\mathbb {H}}_{\mathbb

The recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures

Using the local picture of the degeneration of sequences of minimal surfaces developed by Chodosh et al. (Invent Math 209(3):617–664, [1]) we show that in any closed Riemannian 3-manifold (M, g), the genus of an embedded CMC surface can be bounded only in terms of its index and area, independently of the value of its mean curvature. We also show that if M has finite fundamental group, the genus and

We show that for compact Riemannian manifolds of dimension at least 3 with nonempty boundary, we can modify the manifold by performing surgeries of codimension 2 or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certain changes in the topology of a domain do not have an effect when considering shape optimization questions for Steklov eigenvalues in dimensions 3 and higher

This paper is devoted to the study of dynamical behavior near explicit finite time blowup solutions for three-dimensional incompressible Magnetohydrodynamics (MHD) equations. The global infinite energy solution near a constant vector of three-dimensional incompressible MHD equations without magnetic diffusion has been obtained by Deng–Zhang (Arch Rational Mech Anal 230:1017–1102, 2018). More precisely

In 1939 Weyl showed that the volume of spherical tubes around compact submanifolds M of Euclidean space depends solely on the induced Riemannian metric on M. Can this intrinsic nature of the tube volume be preserved for tubes with more general cross sections \(\mathbb {D}\) than the round ball? Under sufficiently strong symmetry conditions on \(\mathbb {D}\) the answer turns out to be yes.

A deformed Donaldson–Thomas connection for a manifold with a \(\mathrm{Spin}(7)\)-structure, which we call a \(\mathrm{Spin}(7)\)-dDT connection, is a Hermitian connection on a Hermitian line bundle L over a manifold with a \(\mathrm{Spin}(7)\)-structure defined by fully nonlinear PDEs. It was first introduced by Lee and Leung as a mirror object of a Cayley cycle obtained by the real Fourier–Mukai

The aim of this article is to provide a Liouville theorem for heat equation along ancient super Ricci flow. We formulate such a Liouville theorem under a growth condition concerning Perelman’s reduced distance.

This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations extending our findings in Dahlke and Schneider (Anal Appl 17(2):235–291, 2019, Thms. 4.5, 4.9, 4.12, 4.14) to domains of polyhedral type. In particular, we study the smoothness in the specific scale \(\ B^r_{\tau ,\tau }, \ \frac{1}{\tau }=\frac{r}{d}+\frac{1}{p}\ \) of Besov spaces. The regularity

In this paper, we consider a Dirichlet problem driven by an anisotropic (p, q)-differential operator and a parametric reaction having the competing effects of a singular term and of a superlinear perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter moves. Moreover, we prove the existence of a minimal positive solution and determine

We extend the notion of the minimal volume ellipsoid containing a convex body in \(\mathbb {R}^{d}\) to the setting of logarithmically concave functions. We consider a vast class of logarithmically concave functions whose superlevel sets are concentric ellipsoids. For a fixed function from this class, we consider the set of all its “affine” positions. For any log-concave function f on \(\mathbb {R}^{d}

The purpose of this paper is to find pointwise estimates on the integral kernel for the \(\Box _b\) Green operator, N, for a particular quadric submanifold of codimension 2 in \({\mathbb {C}}^4\). This particular kernel has singularities away from the diagonal. Furthermore, the estimates exhibited by this kernel, both on and off the diagonal, follow no known paradigm.

We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces X homeomorphic to \({{\mathbb {R}}}^2\). Given a measure \(\mu \) on such a space, we introduce \(\mu \)-quasiconformal maps \(f:X \rightarrow {{\mathbb {R}}}^2\), whose definition involves deforming lengths of curves by \(\mu \). We show that if \(\mu \) is an infinitesimally metric measure

We introduce the natural notion of (p, q)-harmonic morphisms between Riemannian manifolds. This unifies several theories that have been studied during the last decades. We then study the special case when the maps involved are complex-valued. For these we find a characterisation and provide new non-trivial examples in important cases.

In this paper, we study locally projectively flat Finsler metrics of constant flag curvature. We find equations that characterize these metrics by warped product. Using the obtained equations, we manufacture new locally projectively flat Finsler warped product metrics of vanishing flag curvature. These metrics contain the metric introduced by Berwald and the spherically symmetric metric given by Mo-Zhu

Let f be a harmonic map from a Riemann surface to a Riemannian n-manifold. We prove that if there is a holomorphic diffeomorphism h between open subsets of the surface such that \(f\circ h = f\), then f factors through a holomorphic map onto another Riemann surface. If such h is anti-holomorphic, we obtain an analogous statement. For minimal maps, this result is well known and is a consequence of the

We correct some mistakes in “Entropy in A Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces”.

In this work we obtain sharp \(L^p\)-estimates for pseudo-differential operators on arbitrary graded Lie groups. The results are presented within the setting of the global symbolic calculus on graded Lie groups by using the Fourier analysis associated to every graded Lie group which extends the usual one due to Hörmander on \({\mathbb {R}}^n\). The main result extends the classical Fefferman’s sharp

We consider the existence and regularity of weakly polyharmonic almost complex structures on a compact almost Hermitian manifold \(M^{2m}\). Such objects satisfy the elliptic system \([\varDelta ^m J, J]=0\) weakly. We prove a general regularity theorem for semilinear systems in critical dimensions (with critical growth nonlinearities), which includes the system of polyharmonic almost complex structures

In this paper, we study a broader type of generalized balls which are domains on the complex projective spaces with possibly Levi-degenerate boundaries. We prove rigidity theorems for proper holomorphic mappings among them by exploring the structure of the moduli spaces of projective linear subspaces, which generalize some earlier results for the ordinary generalized balls with Levi-nondegenerate boundaries

We prove that if \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is convex and \(A\subset {\mathbb {R}}^n\) has finite measure, then for any \(\varepsilon >0\) there is a convex function \(g:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) of class \(C^{1,1}\) such that \({\mathcal {L}}^n(\{x\in A:\, f(x)\ne g(x)\})<\varepsilon \). As an application we deduce that if \(W\subset {\mathbb {R}}^n\) is a compact

The Bohr–Fourier series development on one-dimensional solenoids is described using invariant functions and extending Bohr’s theory through the study of transversal variation.

We show that the set of continuous quasinorms on a finite-dimensional linear space, after quotienting by the dilations, has a natural structure of Banach space. Our main result states that, given a finite-dimensional vector space E, the pseudometric defined in the set of continuous quasinorms \(\mathcal {Q}_0=\{\Vert \cdot \Vert :E\rightarrow \mathbb {R}\}\) as $$\begin{aligned} \mathrm{{d}}(\Vert

Non-abelian X-ray tomography seeks to recover a matrix potential \(\Phi :M\rightarrow {\mathbb {C}}^{m\times m}\) in a domain M from measurements of its so-called scattering data \(C_\Phi \) at \(\partial M\). For \(\dim M\ge 3\) (and under appropriate convexity and regularity conditions), injectivity of the forward map \(\Phi \mapsto C_\Phi \) was established in (Paternain et al. in Am J Math 141(6):1707–1750

An inequality is derived for the average energies of pinned distance measures. This refines Mattila’s theorem on distance sets (Mattila in Mathematika 34:207–228, 1987) to pinned distance sets, and gives an analogue of Liu’s theorem (Liu in Geom Funct Anal 29:283–294, 2019) for pinned distance sets of dimension smaller than 1.

We define Hardy spaces \({\mathcal {H}}^p\) for quasiregular mappings in the plane, and show that for a particular class of these mappings many of the classical properties that hold in the classical setting of analytic mappings still hold. This particular class of quasiregular mappings can be characterised in terms of composition operators when the symbol is quasiconformal. Relations between Carleson

In the literature on slice analysis in the hypercomplex setting, there are two main approaches to define slice regular functions in one variable: one consists in requiring that the restriction to any complex plane is holomorphic (with the same complex structure of the complex plane), the second one makes use of stem and slice functions. So far, in the setting of several hypercomplex variables, only

Let \(\Gamma \) be an \(L\)-shape arc consisting of 2 line segments that meet at an angle different from \(\pi \) in the complex \(z\)-plane \({\mathbb C}\). Application of the exterior conformal map \(\psi \) from \(|w| > 1\) onto \({\mathbb C}\backslash \Gamma \), with \(\psi (\infty )= \infty \), introduces the level curves \(\Gamma _n=\{z= \psi (w):|w|=1+{1\over {n+1}}\}\). Let \(\psi ^*\) denote

In this paper, we study the following “slice rigidity property”: given two Kobayashi complete hyperbolic manifolds M, N and a collection of complex geodesics \({\mathscr {F}}\) of M, when is it true that every holomorphic map \(F:M\rightarrow N\) which maps isometrically every complex geodesic of \({\mathscr {F}}\) onto a complex geodesic of N is a biholomorphism? Among other things, we prove that

On any proper convex domain in real projective space there exists a natural Riemannian metric, the Blaschke metric. On the other hand, distances between points can be measured in the Hilbert metric. Using techniques of optimal control, we develop inequalities that provide lower bounds for the Riemannian length of the line segment joining two points of the domain by the Hilbert distance between these

In this note, we find a sharp upper bound for the Steklov spectrum on a submanifold of revolution in Euclidean space with one boundary component.

The star transform is a generalized Radon transform mapping a function of two variables to its integrals along “star-shaped” trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in mathematical models of various imaging modalities based on scattering of elementary particles. The paper presents a comprehensive study of the inversion of the star

Given \(k\in \mathbb {R}\), v, \(D>0\), and \(n\in \mathbb {N}\), let \(\left\{ M_{\alpha }\right\} _{\alpha =1}^{\infty }\) be a Gromov–Hausdorff convergent sequence of Riemannian n–manifolds with sectional curvature \(\ge k\), volume \(>v\), and diameter \(\le D\). Perelman’s Stability Theorem implies that all but finitely many of the \(M_{\alpha }\)s are homeomorphic. The Diffeomorphism Stability

In this article, we establish scale-invariant Strichartz estimates for the Schrödinger equation on arbitrary compact globally symmetric spaces and some bilinear Strichartz estimates on products of rank-one spaces. As applications, we provide local well-posedness results for nonlinear Schrödinger equations on such spaces in both subcritical and critical regularities.

We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of \({\mathbb {R}}^N\). We prove that the symmetric functions of the eigenvalues depend real analytically upon domain perturbations and we prove an Hadamard-type formula for their shape differential. In the case of perturbations depending on a single scalar parameter

In this paper we investigate the property of engulfing for H-convex functions defined on the Heisenberg group \({\mathbb {H}^n}\). Starting from the horizontal sections introduced by Capogna and Maldonado (Proc Am Math Soc 134:3191–3199, 2006) , we consider a new notion of section, called \({\mathbb {H}^n}\)-section, as well as a new condition of engulfing associated to the \({\mathbb {H}^n}\)-sections

In this article we study the local estimates of Ricci flow when the curvature of the initial metric is of almost PIC1 and use it to study complete manifolds with weakly PIC1 and maximal volume growth.

The main purpose of this article is to present a generalization of Forelli’s theorem for the functions holomorphic along a general pencil of holomorphic discs. This generalizes the main result of Joo et al. (Math Ann 355(3):1171–1176, 2013) and the original Forelli’s theorem, and furthermore, answers one of the problems posed in Chirka (Proc Steklov Inst Math 253(2):212–220, 2006).

Given a contact 3-manifold, we consider the problem of when a given function can be realized as the Ricci curvature of a Reeb vector field for the contact structure. We will use topological tools to show that every admissible function can be realized as such Ricci curvature for a singular metric which is an honest compatible metric away from a measure zero set. However, we will see that resolving such

In this paper, we endow the space of continuous translation invariant valuations on convex sets generated by mixed volumes coupled with a suitable Radon measure on tuples of convex bodies with two appropriate norms. This enables us to construct a continuous extension of the convolution operator on smooth valuations to non-smooth valuations, which are in the completion of the spaces of valuations with

For each positive integer d, we prove a uniform \(l^2\)-decoupling inequality for the collection of all polynomials phases of degree at most d. Our result is intimately related to Biswas et al. (Proc Am Math Soc 148(5):1987–1997, 2020), but we use a different partition that is determined by the geometry of each individual phase function.