Related papers: Comments on the del Pezzo cone

It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise…

For each {\it well approximable} irrational $\theta$, we provide an explicit rank-one construction of the $e^{2\pi i\theta}$-rotation $R_\theta$ on the circle $\Bbb T$. This solves "almost surely" a problem by del Junco. For {\it every}…

We introduce a mock toric variety, a generalization of a toric variety. For a non-toric example, Del-Pezzo surfaces are mock toric varieties. These new varieties inherit some properties of mock toric varieties. In application, we give…

We give a new construction of Ricci-flat self-dual metrics which is a natural extension of the Gibbons--Hawking ansatz. We also give characterisations of both these constructions, and explain how they come from harmonic morphisms.

Building upon the classification by Lacini [arXiv:2005.14544], we determine the isomorphism classes of log del Pezzo surfaces of rank one over an algebraically closed field of characteristic five either which are not log liftable over the…

In any dimension $n+1\ge 4$ we construct a sequence of closed $(n+1)$-dimensional Riemannian manifolds with positive Ricci curvature admitting embedded two-sided minimal hypersurfaces such that the following hold: (i) any such hypersurface…

For any elliptic K3 surface $\mathfrak{F}: \mathcal{K} \rightarrow \mathbb{P}^1$, we construct a family of collapsing Ricci-flat K\"ahler metrics such that curvatures are uniformly bounded away from singular fibers, and which…

We prove that the integral points are potentially Zariski dense in the complement of a reduced effective singular anticanonical divisor in a smooth del Pezzo surface, with the exception of $\mathbb{P}^2$ minus three concurrent lines (for…

We investigate Manin's conjecture for del Pezzo surfaces of degree five with a conic bundle structure, proving matching upper and lower bounds, and the full conjecture in the Galois general case.

We discuss whether it is possible to reconstruct a metric by its unparameterized geodesics, and how to do it effectively. We explain why this problem is interesting for general relativity. We show how to understand whether all curves from a…

In this work, we consider a class of Finsler metrics using the warped product notion introduced by Chen, S. and Zhao (2018), with another "warping", one that is consistent with static spacetimes. We will give the PDE characterization for…

We investigate the Kahler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled…

We completely classify redundant blow-ups appearing in the theory of rational surfaces with big anticanonical divisor due to Sakai. In particular, we construct a rational surface with big anticanonical divisor which is not a minimal…

We compute a naturally defined measure of the size of the nef cone of a Del Pezzo surface. The resulting number appears in a conjecture of Manin on the asymptotic behavior of the number of rational points of bounded height on the surface.…

By a classical theorem of Gallot (1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe…

We show that all non-developable ruled surfaces endowed with Ricci metrics in the three-dimensional Euclidean space may be constructed using curves of constant torsion and its binormal. This allows us to give characterizations of the…

In this paper, we give a sufficient condition such that the Ricci flow in $R^2$ exists globally and the flow converges at $t=\infty$ to the flat metric on $R^2$.

We study the concept of cone metric space in the context of ordered vector spaces by setting up a general and natural framework for it.

In the paper we introduce new metric structures on $\mathfrak{g}$-foliations that are less rigid than the well-known structures: almost contact and 3-quasi-Sasakian structures as well as $f$-structures with parallelizable kernel and almost…

In this paper, we study conformal Ricci solitons and conformal gradient Ricci solitons on generalized ($\kappa,\mu$)-space forms. The conditions for the solitons to be shrinking, steady, and expanding are derived in terms of conformal…