Generalized Killing Ricci tensor for real hypersurfaces in the complex quadric

First we introduce a new notion of generalized Killing Ricci tensor which is equivalent to the notion of cyclic parallel Ricci tensor for real hypersurfaces in the complex quadric $Q^m=S{O}_{m+2}/S{O}_{m}S{O}_2$. Next, in terms of $\mathfrak{A}$-principal or $\mathfrak{A}$-isotropic unit normal vector fields we give a complete classification of real hypersurfaces in $Q^m=S{O}_{m+2}/S{O}_{m}S{O}_2$ with cyclic parallel Ricci tensor.     

Quantifying properties (K) and (μs$\mu ^{s}$)

A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence ${(f_n)}_{n=1}^{\infty }$ so that $⟨f_n,x_n⟩\to 0$ as $n\to \infty$ for every weakly null sequence ${(x_n)}_{n=1}^{\infty }$ in X; X has property $({\mu }^s)$ if every weak* null sequence in $X^{\ast }$ admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property $({\mu }^s)$ and reflexivity (or even the Grothendieck property) imply property (K). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.     

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 $2\chi -6\le K^2\le 8\chi -8$.     

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 ${\mathbb{H}}^n\times \mathbb{R}$.     

Totally real flat minimal surfaces in the hyperquadric

In this paper, we study geometry of totally real minimal surfaces in the complex hyperquadric $Q_{N-2}$, and obtain some characterizations of the harmonic sequence generated by these minimal immersions. For totally real flat surfaces that are minimal immersed in both $Q_{N-2}$ and $\mathbb{C}{P}^{N-1}$, we determine them for $N=4,5,6$, and give a classification theorem when they are Clifford solutions.     

Isomorphisms of Galois groups of number fields with restricted ramification

Let K be a number field and S a set of primes of K. We write $K_S/K$ for the maximal extension of K unramified outside S and $G_{K,S}$ for its Galois group. In this paper, we answer the following question under some assumptions: “For $i=1,2$, let $K_i$ be a number field, $S_i$ a (sufficiently large) set of primes of $K_i$ and $\sigma :G_{K_1,S_1}\stackrel{\sim }{\to }{G}_{K_2,S_2}$ an isomorphism. Is σ induced by a unique isomorphism between $K_{1,S_1}/K_1$ and $K_{2,S_2}/K_2$?” Here, the main assumption is about the Dirichlet density of $S_i$.     

On topological and combinatorial structures of pointed stable curves over algebraically closed fields of positive characteristic

In this paper, we study anabelian geometry of curves over algebraically closed fields of positive characteristic. Let $X^{•}=(X,D_X)$ be a pointed stable curve over an algebraically closed field of characteristic $p>0$ and ${\mathrm{\Pi }}_{X^{•}}$ the admissible fundamental group of $X^{•}$. We prove that there exists a group-theoretical algorithm, whose input datum is the admissible fundamental group ${\mathrm{\Pi }}_{X^{•}}$, and whose output data are the topological and the combinatorial structures associated with $X^{•}$. This result can be regarded as a mono-anabelian version of the combinatorial Grothendieck conjecture in the positive characteristic. Moreover, by applying this result, we construct clutching maps for moduli spaces of admissible fundamental groups.     

The rational cuspidal subgroup of J0(p2M)$J_0(p^2M)$ with M squarefree

For a positive integer N, let $X_0(N)$ be the modular curve over $\mathbf{Q}$ and $J_0(N)$ its Jacobian variety. We prove that the rational cuspidal subgroup of $J_0(N)$ is equal to the rational cuspidal divisor class group of $X_0(N)$ when $N=p^{2}M$ for any prime p and any squarefree integer M. To achieve this, we show that all modular units on $X_0(N)$ can be written as products of certain functions $F_{m,h}$, which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on $X_0(N)$ under a mild assumption.     

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.     

A singular Liouville equation on planar domains

We show the existence of a solution for an equation where the nonlinearity is logarithmically singular at the origin, namely, $-\mathrm{\Delta }u=(\log u+f(u)){\chi }_{\{u>0\}}$ in $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with Dirichlet boundary condition. The function f has exponential growth, which can be subcritical or critical with respect to the Trudinger–Moser inequality. We study the energy functional $I_{\epsilon }$ corresponding to the perturbed equation $-\mathrm{\Delta }u+g_{\epsilon }(u)=f(u)$, where $g_{\epsilon }$ is well defined at 0 and approximates $-\log u$. We show that $I_{\epsilon }$ has a critical point $u_{\epsilon }$ in $H_0^1(\mathrm{\Omega })$, which converges to a legitimate nontrivial nonnegative solution of the original problem as $\epsilon \to 0$. We also investigate the problem with $f(u)$ replaced by $\lambda f(u)$, when the parameter $\lambda >0$ is sufficiently large.     

Estimates for the entropy numbers of the Nikol'skii–Besov classes of functions with mixed smoothness in the space of quasi-continuous functions

We obtained order estimates for the entropy numbers of the Nikol'skii–Besov classes of functions $B_{p,\theta }^{\mathit{r}}({\mathbb{T}}^d)$ with mixed smoothness in the metric of the space of quasi-continuous functions $QC({\mathbb{T}}^d)$. We also showed that for $2\le p\le \infty$, , $r_1>\frac{1}{2}$, $d\ge 2$, the estimate of the corresponding asymptotic characteristic is exact in order.     

Exponential decay for semilinear wave equation with localized damping in the hyperbolic space

In this paper, we study the exponential decay of the energy associated to an initial value problem involving the wave equation on the hyperbolic space ${\mathbb{B}}^N$. The space ${\mathbb{B}}^N$ is the unit disc of ${\mathbb{R}}^N$ endowed with the Riemannian metric g given by $g_{ij}=p^2{\delta }_{ij}$, where $p(x)=\frac{2}{1-{|x|}^2}$ and ${\delta }_{ij}=1$, if $i=j$ and ${\delta }_{ij}=0$, if $i\ne j$. Making an appropriate change, the problem can be seen as a singular problem on the boundary of the open ball endowed with the euclidean metric. The proof is based on the multiplier techniques combined with the use of Hardy's inequality, in a version due to the Brezis–Marcus, which allows us to overcome the difficulty involving the singularities.     

On the fixed part of pluricanonical systems for surfaces

We show that $|m{K}_X|$ defines a birational map and has no fixed part for some bounded positive integer m for any $\frac{1}{2}$-lc surface X such that $K_X$ is big and nef. For every positive integer $n\ge 3$, we construct a sequence of projective surfaces $X_{n,i}$, such that $K_{X_{n,i}}$ is ample, $\mathrm{mld}(X_{n,i})>\frac{1}{n}$ for every i, ${\lim }_{i\to +\infty }\mathrm{mld}(X_{n,i})=\frac{1}{n}$, and for any positive integer m, there exists i such that $|m{K}_{X_{n,i}}|$ has nonzero fixed part. These results answer the surface case of a question of Xu.     

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).     

The fractional porous medium equation on manifolds with conical singularities II

This is the second of a series of two papers that studies the fractional porous medium equation, ${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma }{(|u|}^{m-1}u)=0$ with $m>0$ and $\sigma \in (0,1]$, posed on a Riemannian manifold with isolated conical singularities. The first aim of the article is to derive some useful properties for the Mellin–Sobolev spaces including the Rellich–Kondrachov theorem and Sobolev–Poincaré, Nash and Super Poincaré type inequalities. The second part of the article is devoted to the study the Markovian extensions of the conical Laplacian operator and its fractional powers. Then based on the obtained results, we establish existence and uniqueness of a global strong solution for $L_{\infty }-$initial data and all $m>0$. We further investigate a number of properties of the solutions, including comparison principle, $L_p-$contraction and conservation of mass. Our approach is quite general and thus is applicable to a variety of similar problems on manifolds with more general singularities.     

Gauss-Prym maps on Enriques surfaces

We prove that the kth Gaussian map ${\gamma }_H^k$ is surjective on a polarized unnodal Enriques surface $(S,H)$ with $\phi (H)>2k+4$. In particular, as a consequence, when $\phi (H)>4(k+2)$, we obtain the surjectivity of the kth Gauss-Prym map ${\gamma }_{{\omega }_C\otimes \alpha }^k$, with $\alpha :={\omega }_{{S|}_C}$, on smooth hyperplane sections $C\in |H|$. In case $k=1$, it is sufficient to ask $\phi (H)>6$.     

A flow approach to the planar Lp$L_p$ Minkowski problem

This paper will establish the asymptotic behavior of an anisotropic area-preserving flow which shows the existence of smooth solutions of the planar $L_p$ Minkowski problem for $p\ne 0$.