arXiv subjects
Publications and source records attributed to Martin Kilian.
We consider the local theory of constant mean curvature surfaces that satisfy one or two integrable boundary conditions and determine the corresponding potentials for the generalized Weierstrass representation.
A smooth T-surface can be thought of as a generalization of a surface of revolution in such a way that the axis of rotation is not fixed at one point but rather traces a smooth path on the base plane. Furthermore, the action, by which the aforementioned surface is obtained does not need to be merely rotation but any ``suitable" planar equiform transformation applied to the points of a certain smooth profile curve. In analogy to the smooth setting, if the axis footpoints sweep a polyline on the base plane and if the profile curve is discretely chosen then a T-hedra (discrete T-surface) with trapezoidal faces is obtained. The goal of this article is to reconstruct a T-hedron from an already given point cloud of a T-surface. In doing so, a kinematic approach is taken into account, where the algorithm at first tries to find the aforementioned axis direction associated with the point cloud. Then the algorithm finds a polygonal path through which the axis footpoint moves. Finally, by properly cutting the point cloud with the planes passing through the axis and its footpoints, it reconstructs the surface. The presented method is demonstrated on base of examples. From an applied point of view, the straightforwardness of the generation of these surfaces predestines them for building and design processes. In fact, one can find many built objects belonging to the sub-classes of T-surfaces such as \emph{surfaces of revolution} and \emph{moulding surfaces}. Furthermore, the planarity of the faces of the discrete version paves the way for steel/glass construction in industry. Finally, these surfaces are also suitable for transformable designs as they allow an isometric deformation.
We use Whitham deformations to give a complete account of spectral data of real solutions of the sinh--Gordon equation of spectral genus 2. We parameterise the closure of spectral data of constant mean curvature tori in $\mathbb{R}^3$ by an isosceles right triangle and analyse its boundary. We prove that the Wente family, which is described by spectral data with real coefficients, is parameterised by the bisector of the right angle. Our methods combine blowups of Whitham deformations and spectral data in an innovative way that changes the underlying integrable system.
We consider the persistence probability for the integrated fractional Brownian motion and the fractionally integrated Brownian motion with parameter $H,$ respectively. For the integrated fractional Brownian motion, we discuss a conjecture of Molchan and Khokhlov and determine the asymptotic behavior of the persistence exponent as $H\to 0$ and $H\to 1,$ which is in accordance with the conjecture. For the fractionally integrated Brownian motion, also called Riemann-Liouville process, we find the asymptotic behavior of the persistence exponent as $H\to 0$.
We consider the sum of two self-similar centred Gaussian processes with different self-similarity indices. Under non-negativity assumptions of covariance functions and some further minor conditions, we show that the asymptotic behaviour of the persistence probability of the sum is the same as for the single process with the greater self-similarity index. In particular, this covers the mixed fractional Brownian motion introduced in Cheridito (2001) and shows that the corresponding persistence probability decays asymptotically polynomially with persistence exponent $1-\max(1/2,H),$ where $H$ is the Hurst parameter of the underlying fractional Brownian motion.
We study a modification of the fractional analogue of the Brownian meander, which is Brownian motion conditioned to be positive on the time interval ${[0,1]}$. More precisely, we determine the weak limit of a fractional Brownian motion which is penalized -- instead of being killed -- when leaving the positive half-axis. In the Brownian case, we give a representation of the limiting process in terms of an explicit SDE and compare it to the SDE fulfilled by the Brownian meander.
We adapt Sklyanin's $K$-matrix formalism to the sinh-Gordon equation, and prove that all free boundary constant mean curvature (CMC) annuli in the unit ball in $\mathbb{R}^3$ are of finite type.
We show that the spaces of closed finite gap curves in ${\mathbb R}^3$ and ${\mathbb S}^3$ are dense with respect to the Sobolev $W^{2,2}$-norm in the spaces of closed curves in ${\mathbb R}^3$ respectively ${\mathbb S}^3$.
We construct a new five parameter family of constant mean curvature trinoids with two asymptotically Delaunay ends and one irregular end.
We prove that the set of closed finite gap curves in hyperbolic 3-space $\mathbb{H}^3$ is $W^{2,2}$-dense in the Sobolev space of all closed $W^{2,2}$-curves in $\mathbb{H}^3$. We also show that the set of closed finite gap curves in any 2-dimensional space form $\mathbb{E}^2$ is $W^{2,2}$-dense in the Sobolev space of all closed $W^{2,2}$-curves in $\mathbb{E}^2$.
[French] We develop new applications of Sklyanin's $K$-matrix formalism to the study of periodic solutions of the sinh-Gordon equation.
We consider the Whitham equations for deformations of hyperelliptic spectral curves, which preserve all periods of a meromorphic differential. If the meromorphic differential has a root at a fixed point of the hyperelliptic involution, then the Whitham flow has a singularity. We prove that the stable and unstable manifolds are non-empty and extend the Whitham flow continuously through the singularity.
We present a loop group description for curves in $\mathbb{R}^3$, and apply it to classify the circletons: Circles dressed by simple factors.
We consider mean-convex Alexandrov embedded surfaces in the round unit 3-sphere, and show under which conditions it is possible to continuously deform these preserving mean-convex Alexandrov embeddedness.
We present a deformation for constant mean curvature tori in the 3-sphere. We show that the moduli space of equivariant constant mean curvature tori in the 3-sphere is connected, and we classify the minimal, the embedded, and the Alexandrov embedded tori therein. We conclude with an instability result.
We prove the existence of a new class of constant mean curvature cylinders with an arbitrary number of umbilics by unitarizing the monodromy of Hill's equation.
We discuss constant mean curvature bubbletons in Euclidean 3-space via dressing with simple factors, and prove that single bubbletons are not embedded.
We describe infinitesimal deformations of constant mean curvature surfaces of finite type in the 3-sphere. We use Baker-Akhiezer functions to describe such deformations, as well as polynomial Killing fields and the corresponding spectral curve to distinguish between isospectral and non-isospectral deformations.