Related papers: Complete Willmore surfaces in H3 with bounded ener…
We study the Willmore flow for graphs over a bounded domain in $\mathbb{R}^2$ with Dirichlet (clamped) boundary conditions, a still little-studied setting that also serves as a prototype for higher-order flows with fixed boundary data. We…
We investigate the Hawking energy of small surfaces in space times without symmetry assumptions by introducing the notion of Hawking type functionals. In particular, we find that Hawking type functionals are generalized Willmore functionals…
We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,\alpha}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the…
Geometric and topological bounds are obtained for the first energy level gap of a particle constrained to move on a compact surface in 3-space. Moreover, geometric properties are found which allows for stationary and uniformly distributed…
We construct local examples of singular Hermitian Yang-Mills connections over $B_1\subset \mathbb{C}^3$ with uniformly bounded $L^2$-energy, but the number of essential singular points can be arbitrarily large.
We establish an epsilon-regularity theorem at points in the free boundary of almost-minimizers of the energy $\mathrm{Per}_{w}(E)=\int_{\partial^*E}w\,\mathrm{d} {\mathscr{H}}^{n-1}$, where $w$ is a weight asymptotic to…
In this note we study the boundary regularity of minimizers of a family of weak anchoring energies that model the states of liquid crystals. We establish optimal boundary regularity in all dimensions $n\geq 3 .$ In dimension $n=3,$ this…
This work investigates the formation of singularities under the steepest descent $L^2$-gradient flow of the functional $\mathcal W_{\lambda_1, \lambda_2}$, the sum of the Willmore energy, $\lambda_1$ times the area, and $\lambda_2$ times…
This short note is the announcement of a forthcoming work in which we prove a first general boundary regularity result for area-minimizing currents in higher codimension, without any geometric assumption on the boundary, except that it is…
We found a new formulation to the Euler-Lagrange equation of the Willmore functional for immersed surfaces in ${\R}^m$. This new formulation of Willmore equation appears to be of divergence form, moreover, the non-linearities are made of…
We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,\alpha}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the…
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 consider minimization problems of functionals given by the difference between the Willmore functional of a closed surface and its area, when the latter is multiplied by a positive constant weight $\Lambda$ and when the surfaces are…
Consider a sequence of minimal varieties M_i in a Riemannian manifold N such that the boundary measures are uniformly bounded on compact sets. Let Z be the set of points at which the areas of the M_i blow up. We prove that Z behaves in some…
Constant Mean Curvature (CMC) 1-immersions of surfaces into hyperbolic 3-manifolds are natural and yet rather curious objects in hyperbolic geometry with interesting applications. Firstly, Bryant revealed surprising relations between (CMC)…
In this work, singularly perturbed energies arising from discrete $J_1$-$J_3$-models are studied. The energies under consideration consist of a non-convex bulk term and a higher-order regularizing term and are subject to incompatible…
For a bounded smooth domain in the plane and smooth boundary data we consider the minimisation of the Willmore functional for graphs subject to Dirichlet or Navier boundary conditions. For $H^2$-regular graphs we show that bounds for the…
We consider wave maps from $\mathbb R^{2+1}$ to a $C^\infty$-smooth Riemannian manifold, $\mathcal N$. Such maps can exhibit energy concentration, and at points of concentration, it is known that the map (suitably rescaled and translated)…
Consider the dynamics of a gas bubble in an inviscid, compressible liquid with surface tension. Kinematic and dynamic boundary conditions couple the bubble surface deformation dynamics with the dynamics of waves in the fluid. This system…
We give an explicit and complete description of bubbling limits of a non-collapsing limit of polarized K3 surfaces in terms of the period mapping. In particular, we show that bubbling limits only depend on algebro-geometric data of the…