相关论文: A Gap Theorem for Willmore Tori and an application…
We study long-time existence and asymptotic behavior for the $L^2$-gradient flow of the Willmore energy, under the condition that the initial datum is a torus of revolution. We show that if an initial datum has Willmore energy below $8\pi$…
In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in Euclidean three-space is at least 2\pi^2. We prove this conjecture using the min-max theory of minimal surfaces.
In this article, the author investigates flow lines of the classical Willmore flow, which start to move in a smooth parametrization of a Hopf-torus in $\mathbb{S}^3$. We prove that any such flow line of the Willmore flow exists globally, in…
The paper is devoted to study the Dirichelet energy of moving frames on 2-dimensional tori immersed in the euclidean $3\leq m$-dimensional space. This functional, called Frame energy, is naturally linked to the Willmore energy of the…
The Willmore conjecture states that any immersion F:T^2 -> R^n of a 2-torus into flat euclidean space satisfies $\int_{T^2} H^2\geq 2\pi^2$. We prove it under the condition that the L^p-norm of the Gaussian curvature is sufficiently small.
The Willmore Problem seeks the surface in $\mathbb S^3\subset\mathbb R^4$ of a given topological type minimizing the squared-mean-curvature energy $W = \int |\mathbf{H}_{\mathbb{R}^4}|^2 = \operatorname{area} + \int H_{\mathbb{S}^3}^2$. The…
The Willmore energy of a closed surface in R^n is the integral of its squared mean curvature, and is invariant uner M\"obius transformations of R^n. We show that any torus in R^3 with energy at most $8 \pi-delta$ has a representative under…
The Willmore Problem seeks closed surfaces in $\mathbb{S}^3\subset\mathbb{R}^4$ of a given topological type minimizing the squared-mean-curvature energy $W = \int |H_{\mathbb{R}^4}|^2 = area + \int |H_{\mathbb{S}^3}|^2$. The longstanding…
A peculiarity of the geometry of the euclidean 3-sphere $\S3$ is that it allows for the existence of compact without boundary minimally immersed surfaces. Despite a wealthy of examples of such surfaces, the only known tori minimally…
For every two-dimensional torus $T^2$ and every $k\in \mathbb{N}$, $k\ge 3$, we construct a conformal Willmore immersion $f:T^2\to \mathbb{R}^4$ with exactly one point of density $k$ and Willmore energy $4\pi k$. Moreover, we show that the…
The Marques-Neves theorem asserts that among all the torodial (i.e. genus 1) closed surfaces, the Clifford torus has the minimal Willmore energy $\int H^2 \, dA$. % It is a natural conjecture that if one prescribes the isoperimetric Since…
We study surface energies depending on the mean curvature in total spaces of Killing submersions, which extend the classical notion of Willmore energy. Based on a symmetry reduction procedure, we construct vertical tori critical for these…
A family of embedded rotationally symmetric tori in the Euclidean 3-space consisting of two opposite signed constant mean curvature surfaces that converge as varifolds to a double round sphere is constructed. Using complete elliptic…
In this article, by following the method in \cite{PT}, combining Willmore energy with isoperimetric inequalities, we construct two examples of singularities under mean curvature flow in $\mathbb{H}^3$. More precisely, there exists a torus,…
We construct embedded Willmore tori with small area constraint in Riemannian three-manifolds under some curvature condition used to prevent M\"obius degeneration. The construction relies on a Lyapunov-Schmidt reduction; to this aim we…
We extend an energy introduced by Mather to the setting of Almgren-Pitts min-max theory and obtain a parametric, higher-dimensional analogue of Mather's variational barrier theory for twist maps and geodesics on tori. We use this energy to…
Kuwert and Sch\"atzle showed in 2001 that the Willmore flow converges to a standard round sphere, if the initial energy is small. In this situation, we prove stability estimates for the barycenter and the quadratic moment of the surface.…
For an integral $2$-varifold $V\subset \mathbb{S}^3$ with square-integrable mean curvature, unit density, and support of genus at least $1$, assume that its Willmore energy satisfies \[ \mathcal{W}(V)\le 2\pi^2+\delta^2,\qquad…
By the classical Li-Yau inequality, an immersion of a closed surface in $\mathbb{R}^n$ with Willmore energy below $8\pi$ has to be embedded. We discuss analogous results for curves in $\mathbb{R}^2$, involving Euler's elastic energy and…
We show that any embedded minimal torus in S^3 is congruent to the Clifford torus. This answers a question posed by H.B. Lawson, Jr., in 1970.