In this work, we investigate the problem of finding surfaces in the Lorentz-Minkowski 3-space with prescribed skew ($S$) and mean ($H$) curvatures, which are defined through the discriminant of the characteristic polynomial of the shape operator and its trace, respectively. After showing that $H$ and $S$ can be interpreted in terms of the expected value and standard deviation of the normal curvature seen as a random variable, we address the problem of prescribed curvatures for surfaces of revolution. For surfaces with a non-lightlike axis and prescribed $H$, the strategy consists in rewriting the equation for $H$, which is initially a nonlinear second order Ordinary Differential Equation (ODE), as a linear first order ODE with coefficients in a certain ring of hypercomplex numbers along the generating curves: complex numbers for curves on a spacelike plane and Lorentz numbers for curves on a timelike plane. We also solve the problem for surfaces of revolution with a lightlike axis by using a certain ODE with real coefficients. On the other hand, for the skew curvature problem, we rewrite the equation for $S$, which is initially a nonlinear second order ODE, as a linear first order ODE with real coefficients. In all the problems, we are able to find the parameterization for the generating curves in terms of certain integrals of $H$ and $S$.

The purpose of this paper is to introduce a notion of pairs of adelic $\mathbb{R}$-Cartier divisors and $\mathbb{R}$-base conditions, to define the arithmetic volumes of such pairs, and to establish fundamental positivity properties of such pairs. We show that the arithmetic volume of a pair has the Fujita approximation property and that the Gâteaux derivatives of the arithmetic volume function at a big pair along the directions of adelic $\mathbb{R}$-Cartier divisors are given by suitable arithmetic positive intersection numbers. As a consequence, we establish an Arakelov theoretic analogue of the classical Bonnesen--Diskant inequality in convex geometry.

We study $L^p$-boundedness of commutators of Bochner-Riesz operators for elliptic self-adjoint operators which satisfy the finite speed of propagation property for the corresponding wave equation. Our results can be applied to Schrödinger operators with inverse square potentials on $\mathbb{R}^n$, elliptic operators on compact manifolds, and Schrödinger operators on asymptotically conic manifolds

Our proof is new even for the commutator of the classical Bochner-Riesz operator when $L$ is the Laplace operator $\Delta=\sum_{i=1}^n\partial_{x_i}^2$ on the Euclidean space $\mathbb{R}^n$.

In this paper, we first prove the $f$-mean curvature comparison in a smooth metric measure space when the Bakry--Emery Ricci tensor is bounded below and $f$ is bounded by a linear function of distance. Based on this, we obtain Myers-type compactness theorems by generalizing the results of Cheeger, Gromov, and Taylor and Wan to the Bakry--Emery Ricci tensor. Moreover, we improve a result of Soylu by using a weaker condition on a derivative of $f$.

We compare three different characterizations, due respectively to R. Dedekind, K. Uchida, and H. Lüneburg, of when, given an algebraic integer, the ring generated by it coincides with the ring of integers of its quotient field, and apply our results to some concrete 2-towers of number fields.

We prove that the higher Nash blowup of a normal toric variety defined over a field of positive characteristic is an isomorphism if and only if it is non-singular. We also extend a result by R. Toh-Yama which shows that higher Nash blowups do not give a one-step resolution of certain toric surface. These results were previously known only in characteristic zero.

Let $M$ be a connected complete noncompact $n$-dimensional Riemannian manifold with a base point $p \in M$ whose radial sectional curvature at $p$ is bounded from below by that of a noncompact surface of revolution which admits a finite total curvature where $n \ge 2$. Note here that our radial curvatures can change signs wildly. We then show that $\lim_{t\to\infty}\mathrm{vol} B_t(p) / t^n$ exists where $\mathrm{vol} B_t(p)$ denotes the volume of the open metric ball $B_t(p)$ with center $p$ and radius $t$. Moreover we show that in addition if the limit above is positive, then $M$ has finite topological type and there is therefore a finitely upper bound on the number of ends of $M$.