We give a Gauss-Bonnet type theorem for future directed fronts in hyperbolic 3-space and de Sitter 3-space. This theorem relates the lightlike geometry to the topology of these singular objects.

In this paper we study properties of complex plane projective curves from a geometric point of view. We focus our attention on properties of conics. We find that Gauss curvature determines a conic up to a hermitian transformation preserving the Fubini-Study metric of the complex projective plane and we discuss some other geometric properties of the conics. Finally we study the deformation of smooth conics onto pair of lines and the classification of cubics up to hermitian transformations.

We investigate the relation between the level set approach and the varifold approach for the mean curvature flow with Neumann boundary conditions. With an appropriate initial data, we prove that the almost all level sets of the unique viscosity level set solution satisfy Brakke's inequality and a generalized Neumann boundary condition.

In this note, we prove the cofiniteness of local cohomology modules $H^i_I(N)$ with respect to $I$ for all $i\lt t$ and the finiteness of $(0:_{H^t_I(N)}I)$ and ${\rm Ext}^1_R(R/I,H^t_I(N))$ provided that ${\rm Ext}^i_R(R/I,N)$ is finitely generated for all $i\le t+1$ and $H^i_I(N)$ is in dimension $\lt 2$ for all $i \lt t$, where $t\ge 1$ is an integer (here, $N$ is not necessarily finitely generated over $R$). This extends the results of Bahmanpour-Naghipour [5, Theorem 2.6], Aghapournahr-Bahmanpour [2, Theorem 3.4], Bahmanpour-Naghipour-Sedghi [4, Theorem 2.8] by a different proof method.

We study a family of mappings from the powers of the unit tangent sphere at a point to a complete Riemannian manifold with non-positive sectional curvature, whose behavior is related to the spherical mean operator and the geodesic random walks on the manifold.

We show that for odd powers of the unit tangent sphere the mappings are fold maps.

Some consequences on the regularity of the transition density of geodesic random walks, and on the eigenfunctions of the spherical mean operator are discussed and related to previous work.

We give a description of Nurowski's conformal structure for some examples of bracket-generating rank 2 distributions in dimension 5, aka $(2,3,5)$-distributions, namely the An-Nurowski circle twistor distribution for pairs of surfaces of constant Gauss curvature rolling without slipping or twisting over each other. In the case of hyperboloid surfaces whose curvature ratios give maximally symmetric $(2,3,5)$-distributions, we find the change of coordinates that map the conformal structure to the flat metric of Engel. We also consider a rank $3$ Pfaffian system in dimension 5 with $SU(2)$ symmetry obtained by rotating two of the 1-forms in the Pfaffian system of the spheres rolling distribution, and discuss complexifications of such distributions.

We consider the singular limit problem in a real Hilbert space for abstract second order evolution equations with a parameter $\varepsilon \in (0,1]$. We first give an alternative proof of the celebrated results due to Kisynski [10] from the viewpoint of the energy method. Next we derive a more precise asymptotic profile as $\varepsilon \to +0$ of the solution itself depending on $\varepsilon$ under a suitable regularity of the initial data.

We show that one can construct a Cartan connection on the Cartan principal bundle over a contact Riemannian manifold, on which the associated complex structure is not assumed to be integrable, according to Cartan-Chern-Moser-Le's construction but with the use of the Tanno connection (instead of the Tanaka-Webster connection in the integrable case). Then we prove that it is normal in the sense of Tanaka if and only if the complex structure is integrable. By Le this has been shown to hold true in the case the dimension of the manifold is three.