We obtain sharp two-sided heat kernel estimates for some process whose regular Dirichlet form is strongly local on spaces with varying dimension, in which two spaces of general dimension are connected at one point. On these spaces, if the dimensions of the two constituent parts are different, the volume doubling property fails with respect to the measure induced by the associated Lebesgue measures. Thus the parabolic Harnack inequalities fail and the heat kernels do not enjoy Aronson type estimates. Our estimates show that the on-diagonal estimates are independent of the dimensions of the two parts of the space for small time, whereas they depend on their transience or recurrence for large time. These are multidimensional version of a space considered by Z.-Q. Chen and S. Lou (Ann. Probab. 2019), in which a 1-dimensional space and a 2-dimensional space are connected at one point.

Let $f: S\to B$ be a surface fibration of genus $g\ge 2$ over ${\mathbb{C}}$. The semistable reduction theorem asserts there is a finite base change $\pi: B'\to B$ such that the fibration $S\times_BB'\to B'$ admits a semistable model. An interesting invariant of $f$, denoted by $N(f)$, is the minimum of $\deg(\pi)$ for all such $\pi$. In an early paper of Xiao, he gives a uniform multiplicative upper bound $N_g$ for $N(f)$ depending only on the fibre genus $g$. However, it is not known whether Xiao's bound is sharp or not. In this paper, we give another uniform upper bound $N'_g$ for $N(f)$ when $f$ is hyperelliptic. Our $N'_g$ is optimal in the sense that for every $g\ge 2$ there is a hyperelliptic fibration $f$ of genus $g$ so that $N(f)=N_g'$. In particular, Xiao's upper bound $N_g$ is optimal when $N_g=N'_g$. We show that this last equation $N_g=N_g'$ holds for infinitely many $g$.

In this note, we investigate local isometric immersion of Clairaut metrics in Euclidean three-space. A Clairaut metric is determined up to isometry by a single function of one variable. We show that an isometric immersion is formally determined by two functions of one variable, uniquely up to coordinate reflection and ambient Euclidean motions. Further, if the Clairaut metric and these two functions are real-analytic, there exists a local isometric immersion realizing these data. We give a more explicit description for a finite-dimensional family of examples.

Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we introduce a subdivision method by applying the Goldberg-Coxeter subdivision and discuss the convergence of a sequence of discrete surfaces defined inductively by the subdivision. We also study the limit set as the continuum geometric object associated with the given discrete surface.

We suggest two versions of the Hardy--Littlewood--Sobolev inequality for discrete time martingales. In one version, the fractional integration operator is a martingale transform, however, it may vanish if the filtration is excessively irregular; the second version lacks the martingale property while being analytically meaningful for an arbitrary filtration.

Consider the Lorentz-Minkowski 3-space ${\mathbb L}^3$ with the metric $dx^2+dy^2-dz^2$ in canonical coordinates $(x,y,z)$. A surface in ${\mathbb L}^3$ is said to be separable if it satisfies an equation of the form $f(x)+g(y)+h(z)=0$ for some smooth functions $f$, $g$ and $h$ defined in open intervals of the real line. In this article we classify all zero mean curvature surfaces of separable type, providing a method of construction of examples.

Our aim in this paper is to establish Hardy-Sobolev inequality in the settings of Herz spaces. As an application, we show Sobolev-type integral representation for a $C^1$-function on ${\mathbb R}^N \setminus \{0\}$ which vanishes outside the unit ball.

The space of multi-vector fields on a manifold has a structure of ${\mathbb{Z}}$-graded Lie superalgebras under the Schouten bracket. (We call such an algebra a pre Lie superalgebra.) As in the case of Lie algebras, we have the associated homology groups similarly. We treat the case where the underlying manifold is a Euclidean $n$-space, and the coefficients are polynomials. Under such conditions, we show the vanishing of the second homology group of the chain complexes which are doubly indexed by the pairs of integers $(w,h)$, which we call weights. We specially treat the case $w=h$ in this article.

There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable $\mathcal{G}_1,\dots,\mathcal{G}_6$ and four are non-orientable $\mathcal{B}_1,\dots,\mathcal{B}_4$. In the present paper we investigate the manifold $\mathcal{G}_6$, also known as Hantzsche-Wendt manifold; this is the unique Euclidean $3$-form with finite first homology group $H_1(\mathcal{G}_6) = \mathbb{Z}^2_4$.The aim of this paper is to describe all types of $n$-fold coverings over $\mathcal{G}_{6}$ and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental group $\pi_1(\mathcal{G}_{6})$ up to isomorphism. Given index $n$, we calculate the numbers of subgroups and the numbers of conjugacy classes of subgroups for each isomorphism type and provide the Dirichlet generating series for the above sequences.