Weighted fractional Sobolev spaces as interpolation spaces in bounded domains

We characterize the real interpolation space between a weighted $L^p$ space and a weighted Sobolev space in arbitrary bounded domains in ${\mathbb{R}}^n$, with weights that are positive powers of the distance to the boundary.     

Elliptic Weingarten hypersurfaces of Riemannian products

Let $M^n$ be either a simply connected space form or a rank-one symmetric space of the noncompact type. We consider Weingarten hypersurfaces of $M\times \mathbb{R}$, which are those whose principal curvatures $k_1,\cdots ,k_n$ and angle function $𝛩$ satisfy a relation $W(k_1,\cdots ,k_n,{𝛩}^2)=0$, being W a differentiable function which is symmetric with respect to $k_1,\cdots ,k_n$. When $\partial W/\partial k_i>0$ on the positive cone of ${\mathbb{R}}^n$, a strictly convex Weingarten hypersurface determined by W is said to be elliptic. We show that, for a certain class of Weingarten functions W, there exist rotational strictly convex Weingarten hypersurfaces of $M\times \mathbb{R}$ which are either topological spheres or entire graphs over M. We establish a Jellett–Liebmann-type theorem by showing that a compact, connected and elliptic Weingarten hypersurface of either ${\mathbb{S}}^n\times \mathbb{R}$ or ${\mathbb{H}}^n\times \mathbb{R}$ is a rotational embedded sphere. Other uniqueness results for complete elliptic Weingarten hypersurfaces of these ambient spaces are obtained. We also obtain existence results for constant scalar curvature hypersurfaces of ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$ which are either rotational or invariant by translations (parabolic or hyperbolic). We apply our methods to give new proofs of the main results by Manfio and Tojeiro on the classification of constant sectional curvature hypersurfaces of ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$.     

Self-repulsiveness of energies for closed submanifolds

We show that the regularized Riesz α-energy for smooth closed submanifolds M in ${\mathbb{R}}^n$ blows up as M degenerates to have double points if $\alpha \le -2dimM$. This gives theoretical foundation of numerical experiments to evolve surfaces to decrease the energy that have been carried out since the 90's.     

Vanishing of higher order Alexander-type invariants of plane curves

The higher order degrees are Alexander-type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve C given as a transversal union of plane curves $C^{\prime }$ and $C^{\prime \prime }$ in terms of the finiteness and the vanishing properties of the invariants of $C^{\prime }$ and $C^{\prime \prime }$, and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial ${\mathrm{\Delta }}_C^{multi}$ is a power of $(t-1)$, and we characterize when ${\mathrm{\Delta }}_C^{multi}=1$ in terms of the defining equations of $C^{\prime }$ and $C^{\prime \prime }$. Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.     

Theorems of Wolff–Denjoy type for semigroups of nonexpansive mappings in geodesic spaces

We show that if $S={\{f_t:Y\to Y\}}_{t\ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings acting on a complete locally compact geodesic space $(Y,d)$ that satisfies some geometric properties, then there exists $\xi \in \partial Y$ such that S converge uniformly on bounded sets of Y to ξ. In particular, our result applies to strictly convex bounded domains in ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$ with respect to a large class of metrics including Hilbert's and Kobayashi's metrics.     

Dimension-free square function estimates for Dunkl operators

Dunkl operators may be regarded as differential-difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood–Paley square function for Dunkl heat flows in ${\mathbb{R}}^d$ is introduced by employing the full “gradient” induced by the corresponding carré du champ operator and then the $L^p$ boundedness is studied for all $p\in (1,\infty )$. For $p\in (1,2]$, we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the $L^p$ boundedness, while for $p\in [2,\infty )$, we restrict to a particular case when the corresponding Weyl group is isomorphic to ${\mathbb{Z}}_2^d$ and apply a probabilistic method to prove the $L^p$ boundedness. In the latter case, the curvature-dimension inequality for Dunkl operators in the sense of Bakry–Emery, which may be of independent interest, plays a crucial role. The results are dimension-free.     

Thom property and Milnor–Lê fibration for analytic maps

Let (X, 0) be the germ of either a subanalytic set $X\subset {\mathbb{R}}^n$ or a complex analytic space $X\subset {\mathbb{C}}^n$, and let $f:(X,0)\to ({\mathbb{K}}^k,0)$ be a $\mathbb{K}$-analytic map-germ, with $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, respectively. When $k=1$, there is a well-known topological locally trivial fibration associated with f, called the Milnor–Lê fibration of f, which is one of the main pillars in the study of singularities of maps and spaces. However, when $k>1$ that is not always the case. In this paper, we give conditions which guarantee that the image of f is well-defined as a set-germ, and that f admits a Milnor–Lê fibration. We also give conditions for f to have the Thom property. Finally, we apply our results to mixed function-germs of type $f\overline{g}:(X,0)\to (\mathbb{C},0)$ on a complex analytic surface $X\subset {\mathbb{C}}^n$ with arbitrary singularity.     

Spectral heat content on a class of fractal sets for subordinate killed Brownian motions

We study the spectral heat content for a class of open sets with fractal boundaries determined by similitudes in ${\mathbb{R}}^d$, $d\ge 1$, with respect to subordinate killed Brownian motions via $\alpha /2$-stable subordinators and establish the asymptotic behavior of the spectral heat content as $t\to 0$ for the full range of $\alpha \in (0,2)$. Our main theorems show that these asymptotic behaviors depend on whether the sequence of logarithms of the coefficients of the similitudes is arithmetic when $\alpha \in [d-\mathfrak{b},2)$, where $\mathfrak{b}$ is the interior Minkowski dimension of the boundary of the open set. The main tools for proving the theorems are the previous results on the spectral heat content for Brownian motions and the renewal theorem.     

Invariant hypercomplex structures and algebraic curves

We show that $U(k)$-invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in $\mathfrak{g}\mathfrak{l}(k,\mathbb{C})$ correspond to algebraic curves C of genus ${(k-1)}^2$, equipped with a flat projection $\pi :C\to {\mathbb{P}}^1$ of degree k, and an antiholomorphic involution $\sigma :C\to C$ covering the antipodal map on ${\mathbb{P}}^1$.     

Geography of minimal surfaces of general type with Z22$\mathbb {Z}_2^2$-actions and the locus of Gorenstein stable surfaces

In this note, the geography of minimal surfaces of general type admitting ${\mathbb{Z}}_2^2$-actions is studied. More precisely, it is shown that Gieseker's moduli space ${\mathfrak{M}}_{K^2,\chi }$ contains surfaces admitting a ${\mathbb{Z}}_2^2$-action for every admissible pair $(K^2,\chi )$ such that $2\chi -6\le K^2\le 8\chi -8$ or $K^2=8\chi$. The examples considered allow to prove that the locus of Gorenstein stable surfaces is not closed in the KSBA-compactification ${\overline{\mathfrak{M}}}_{K^2,\chi }$ of Gieseker's moduli space ${\mathfrak{M}}_{K^2,\chi }$ for every admissible pair $(K^2,\chi )$ such that $2\chi -6\le K^2\le 8\chi -8$.     

Logarithmically improved extension criteria involving the pressure for the Navier–Stokes equations in Rn$\mathbb {R}^{n}$

We consider the Cauchy problem of the Navier–Stokes equations in ${\mathbb{R}}^n$ ($n\ge 3)$ and establish several new extension criteria involving the pressure or its gradient. In particular, we improve the previous results by means of the homogeneous Besov space with negative differential orders in the case $n=3$. Our method is based on the interpolation inequality and the trilinear estimate due to Gérard–Meyer–Oru (Séminaire É. D. P. (1996–1997), Exp. No. IV, 1–8) and Guo–Kučera–Skalák (J. Math. Anal. Appl. 458 (2018) 755–766), respectively.     

Weyl families of transformed boundary pairs

Let $(\mathfrak{L},\mathrm{\Gamma })$ be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space $\mathfrak{H}$. Let $M_{\mathrm{\Gamma }}$ be the Weyl family corresponding to $(\mathfrak{L},\mathrm{\Gamma })$. We cope with two main topics. First, since $M_{\mathrm{\Gamma }}$ need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation $M_{\mathrm{\Gamma }}(z)$, for some $z\in \mathbb{C}\setminus \mathbb{R}$, becomes a nontrivial task. Regarding $M_{\mathrm{\Gamma }}(z)$ as the (Shmul'yan) transform of $zI$ induced by Γ, we give conditions for the equality in $\overline{M_{\mathrm{\Gamma }}(z)}\subseteq \overline{M_{\overline{\mathrm{\Gamma }}}(z)}$ to hold and we compute the adjoint $M_{\overline{\mathrm{\Gamma }}}{(z)}^{\ast }$. As an application, we ask when the resolvent set of the main transform associated with a unitary boundary pair for $T^{+}$ is nonempty. Based on the criterion for the closeness of $M_{\mathrm{\Gamma }}(z)$, we give a sufficient condition for the answer. From this result it follows, for example, that, if T is a standard linear relation in a Pontryagin space, then the Weyl family $M_{\mathrm{\Gamma }}$ corresponding to a boundary relation Γ for $T^{+}$ is a generalized Nevanlinna family; a similar conclusion is already known if T is an operator. In the second topic, we characterize the transformed boundary pair $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$ with its Weyl family $M_{{\mathrm{\Gamma }}^{\prime }}$. The transformation scheme is either ${\mathrm{\Gamma }}^{\prime }=\mathrm{\Gamma }{V}^{-1}$ or ${\mathrm{\Gamma }}^{\prime }=V\mathrm{\Gamma }$ with suitable linear relations V. Results in this direction include but are not limited to: a 1-1 correspondence between $(\mathfrak{L},\mathrm{\Gamma })$ and $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$; the formula for $M_{{\mathrm{\Gamma }}^{\prime }}-M_{\mathrm{\Gamma }}$, for an ordinary boundary triple and a standard unitary operator V (first scheme); construction of a quasi boundary triple from an isometric boundary triple $(\mathfrak{L},{\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_1)$ with $ker\mathrm{\Gamma }=T$ and $T_0=T_0^{\ast }$ (second scheme, Hilbert space case).