Related papers: A support theorem for SLE curves

We study the topology of $SLE$ curves for $\kappa > 4$. More precisely, we show that, a.s., there is no homeomorphism $\Phi: \overline{\mathbb{H}} \rightarrow \overline{\mathbb{H}}$, taking the range of one independent $SLE$ curve to…

When studying stochastic processes, it is often fruitful to have an understanding of several different notions of regularity. One such notion is the optimal H\"older exponent obtainable under reparametrization. In this paper, we show that…

Given a normed plane $\mathcal{P}$, we call $\mathcal{P}$-cycloids the planar curves which are homothetic to their double $\mathcal{P}$-evolutes. It turns out that the radius of curvature and the support function of a $\mathcal{P}$-cycloid…

Whole-plane SLE$_\kappa$ is a random fractal curve between two points on the Riemann sphere. Zhan established for $\kappa \leq 4$ that whole-plane SLE$_\kappa$ is reversible, meaning invariant in law under conformal automorphisms swapping…

The Green's function for the chordal Schramm-Loewner evolution $SLE_\kappa$ for $0 < \kappa < 8$, gives the normalized probability of getting near points. We give up-to-constant bounds for the two-point Green's function.

Let f(x) belong to L^p(R^n) and R>0. The transform is considered that integrates the function f over (almost) all spheres of radius R in R^n. This operator is known to be non-injective (as one can see by taking Fourier transform). However,…

We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle given in terms of the exponential of Gaussian Free Field. We conjecture that our curves are locally…

We consider the Schramm-Loewner evolution (SLE$_\kappa$) for $\kappa \in (4,8)$, which is the regime where the curve is self-intersecting but not space-filling. We show that there exists $\delta_0>0$ such that for $\kappa \in (8 -…

For $E/K$ an elliptic curve without complex multiplication we bound the index of the image of $\operatorname{Gal}(\bar{K}/K)$ in $\operatorname{GL}_2(\hat{\mathbb{Z}})$, the representation being given by the action on the Tate modules of…

The present paper is concerned with properties of multiple Schramm--Loewner evolutions (SLEs) labelled by a parameter $\kappa\in (0,8]$. Specifically, we consider the solution of the multiple Loewner equation driven by a time change of…

Stochastic Loewner Evolution (SLE_kappa) has been introduced as a description of the continuum limit of cluster boundaries in two-dimensional critical systems. We show that the problem of N radial SLEs in the unit disc is equivalent to…

Let $X$, $D$ be a smooth projective surface and a simple normal crossing divisor on $X$, respectively. Suppose $\kappa (X, K_X + D)\ge 0$, let $C$ be an irreducible curve on $X$ whose support is not contained in $D$ and $\alpha$ a rational…

In this note we prove an effective characterization of when two finite-degree covers of a connected, orientable surface of negative Euler characteristic are isomorphic in terms of which curves have simple elevations, weakening the…

Let $\mathcal{P}_{\kappa_1}^{\kappa_2}(\boldsymbol{P}, \boldsymbol{Q})$ denote the set of $C^1$ regular curves in the $2$-sphere $\mathbb{S}^2$ that start and end at given points with the corresponding Frenet frames $\boldsymbol{P}$ and…

Loewner driving functions encode simple curves in 2-dimensional simply connected domains by real-valued functions. We prove that the Loewner driving function of a $C^{1,\beta}$ curve (differentiable parametrization with $\beta$-H\"older…

For $\kappa\in(0,4]$, a family of annulus SLE$(\kappa;\Lambda)$ processes were introduced in [14] to prove the reversibility of whole-plane SLE$(\kappa)$. In this paper we prove that those annulus SLE$(\kappa;\Lambda)$ processes satisfy a…

Given two finite covers $p: X \to S$ and $q: Y \to S$ of a connected, oriented, closed surface $S$ of genus at least $2$, we attempt to characterize the equivalence of $p$ and $q$ in terms of which curves lift to simple curves. Using…

In this paper we define and prove of the existence of the multi-point Green's function for SLE - a normalized limit of the probability that an $SLE_{\kappa}$ curve passes near to a pair of marked points in the interior of a domain. When…

We prove the existence and uniqueness of multiple SLE$_\kappa$ associated with any given link pattern for $\kappa\in (4,6]$. We also have the uniqueness for $\kappa\in (6,8)$. The multiple SLE$_\kappa$ law is constructed by first…

A set of equations is developed to describe a curve in space given the curvature $\kappa$ and the angle of rotation $\theta$ of the osculating plane. The set of equations has a solution (in terms of $\kappa$ and $\theta$) that indirectly…