Related papers: Existence and Regularity for a Curvature Dependent…

In this article, we prove that there exists a unique perimeter minimizer among all piecewise smooth simple closed curves in $M_{\kappa}^2$ enclosing area $A > 0$ $(A \leq 2{\pi}$ if ${\kappa} = 1)$, and it is a circle in $M_{\kappa}^2$ of…

In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional $\int \sqrt{1+K_\gamma^2} ds$, depending both on length and curvature $K$. We fix starting and ending points…

We establish necessary conditions for a regular curve to lie on a circular cylinder in terms of its curvature $\kappa$ and torsion $\tau$. By identifying a fundamental function $\psi = \sin^2 \alpha$, representing the squared sine of the…

We develop an analytic theory of existence and regularity of surfaces (given by graphs) arising from the geometric minimization problem $$\min_{\mathcal{M}}\frac{1}{2}\int_{\mathcal{M}}|\nabla_{\mathcal{M}}H|^2\,dA$$ where $\mathcal{M}$…

Let C be a smooth closed curve of length 2 Pi in R^3, and let k(s) be its curvature, regarded as a function of arc length. We associate with this curve the one-dimensional Schrodinger operator H_C = -d^2/ds^2 + k^2 acting on the space of…

Using Weitzenb\"ock techniques on any compact Riemannian spin manifold we derive inequalities that involve a real parameter and join the eigenvalues of the Dirac operator with curvature terms. The discussion of these inequalities yields…

We estimate from below by geometric data the eigenvalues of the periodic Sturm-Liouville operator $- 4 d^2/ds^2 + \kappa^2 (s)$ with potential given by the curvature of a closed curve.

Motivated by homothetic solutions to curvature-driven flows of planar curves, as well as their many physical applications, this work carries out a systematic study of oriented curves whose curvature $\kappa$ is a given function of position…

In this paper we use a gradient flow to deform closed planar curves to curves with least variation of geodesic curvature in the $L^2$ sense. Given a smooth initial curve we show that the solution to the flow exists for all time and,…

We consider the Dirichlet Laplacian in unbounded strips on ruled surfaces in any space dimension. We locate the essential spectrum under the condition that the strip is asymptotically flat. If the Gauss curvature of the strip equals zero,…

We provide a purely variational proof of the existence of eigenvalues below the bottom of the essential spectrum for the Schr\"odinger operator with an attractive $\delta$-potential supported by a star graph, i.e. by a finite union of rays…

We study the problem of finding curves of minimum pointwise-maximum arc-length derivative of curvature, here simply called curves of minimax spirality, among planar curves of fixed length with prescribed endpoints and tangents at the…

The development of Schramm--Loewner evolution (SLE) as the scaling limits of discrete models from statistical physics makes direct simulation of SLE an important task. The most common method, suggested by Marshall and Rohde \cite{MR05}, is…

We establish sufficient conditions for existence of curves minimizing length as measured with respect to a degenerate metric on the plane while enclosing a specified amount of Euclidean area. Non-existence of minimizers can occur and…

Among all metrics on $\mathbb S^d$ with $d>4$ that are conformal to the standard metric and have positive scalar curvature, the total $\sigma_2$-curvature, normalized by the volume, is uniquely (up to M\"obius transformations) minimized by…

We give a necessary and suficente condition for the existence of a space curve with curvature $\kappa$ and torsion $\tau$ finding a solution of a nonlinear differential equation of second order and some applications are given for the…

We investigate the Schur-complement curvature of D_N-equivariant folded exponential families on the simplex. Our main structural results are: (i) the curvature kappa_Schur(theta) is convex in the log-parameter theta = ln(q); (ii) it admits…

We establish existence and uniqueness results for the modified binormal curvature flow equation that generalizes the binormal curvature flow equation for a curve in $\R^3.$ In this generalization, the velocity of the curve is still directed…

This paper is devoted to the study of shape optimization problems for the first eigenvalue of the elliptic operator with drift L = --$\Delta$+V (x)\cdot \nabla with Dirichlet boundary conditions, where V is a bounded vector field. In 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…