arXiv research

Bi-Lipschitz proof for 2-varifolds near critical Allard condition.

Sharp criteria for 2-varifolds to be induced by smooth immersions.

The paper proves surfaces close to spheres under specific conditions.

Given a Hermitian line bundle $L\to M$ over a closed, oriented Riemannian manifold $M$, we study the asymptotic behavior, as $ε\to 0$, of couples $(u_ε,\nabla_ε)$ critical for the rescalings \begin{align*} &E_ε(u,\nabla)=\int_M\Big(|\nabla u|^2+ε^2|F_\nabla|^2+\frac{1}{4ε^2}(1-|u|^2)^2\Big) \end{align*} of the self-dua…

We use min-max techniques to produce nontrivial solutions $u_ε:M\to \mathbb{R}^2$ of the Ginzburg-Landau equation $Δu_ε+\frac{1}{ε^2}(1-|u_ε|^2)u_ε=0$ on a given compact Riemannian manifold, whose energy grows like $|\logε|$ as $ε\to 0$. When the degree one cohomology $H^1_{dR}(M)=0$, we show that the energy of these s…

Generalizes Reilly inequality to varifolds and analyzes equality cases.

We develop a suitable generalization of Almgren's theory of varifolds in a lorentzian setting, focusing on area, first variation, rectifiability, compactness and closure issues. Motivated by the asymptotic behaviour of the scaled hyperbolic Ginzburg-Landau equations, and by the presence of singularities in lorentzian m…

The paper proves rigidity estimates for varifolds almost minimizing the Willmore energy.

The Clifford torus minimizes Willmore energy closely for small perturbations.

We establish a new estimate for the Ginzburg-Landau energies $E_ε(u)=\int_M\frac{1}{2}|du|^2+\frac{1}{4ε^2}(1-|u|^2)^2$ of complex-valued maps $u$ on a compact, oriented manifold $M$ with $b_1(M)\neq 0$, obtained by decomposing the harmonic component $h_u$ of the one-form $ju:=u^1du^2-u^2du^1$ into an integral and frac…

On a compact manifold $M^{n}$ ($n\geq 3$) with boundary, we study the asymptotic behavior as $ε$ tends to zero of solutions $u_ε: M \to \mathbb{C}$ to the equation $Δu_ε + ε^{-2}(1 - |u_ε|^{2})u_ε = 0$ with the boundary condition $\partial_νu_ε = 0$ on $\partial M$. Assuming an energy upper bound on the solutions and a…

The paper studies surfaces in a bounded domain with orthogonal boundaries and proves curvature estimates.

We study the asymptotics as $p\uparrow 2$ of stationary $p$-harmonic maps $u_p\in W^{1,p}(M,S^1)$ from a compact manifold $M^n$ to $S^1$, satisfying the natural energy growth condition $\int_M|du_p|^p=O(\frac{1}{2-p}).$ Along a subsequence $p_j\to 2$, we show that the singular sets $Sing(u_{p_j})$ converge to the sup…

For Ginzburg-Landau vortices, energy quantization holds only when density is less than 2.

The paper studies the behavior of Möbius-invariant Willmore flow in 3-sphere, proving convergence to Clifford torus.