No compromise in the liquid drop model
We characterize minimizers in Gamow's liquid drop problem.
math.DG↗
arXiv subjects
Matilde Gianocca
Publications and source records attributed to Matilde Gianocca.
Minimal hypersurfaces of Morse index one
We prove that a complete, connected, embedded, minimal hypersurface in $\mathbb{R}^{n+1}$ with finite total curvature and Morse index one is the higher-dimensional catenoid.
math.DG↗
Note on energy index and first eigenvalue of minimal surfaces in spheres
A minimal immersion from a surface to $S^3$ can be viewed both as a critical point of the area and of the energy. Although no difference appears at first order, looking at the respective second variations unveils significant differences. It is well known that whenever the first eigenvalue satisfies $\lambda_1(\Sigma)\geq2$, the index is $\mathrm{ind}_E(\Sigma)\leq 4$. The converse implication is much more subtle. We prove that whenever $\lambda_1(\Sigma)<\frac{1}{6}$, there exists a vector field $X$, orthogonal to the four M\"obius vector fields, with negative second variation. We also prove an arbitrary codimension version of this statement: any immersed minimal surface $\Sigma\subset S^n$ with first eigenvalue $\lambda_1(\Sigma)<\frac{n-2}{2n}$ admits a vector field $X$ orthogonal to the $n+1$ M\"obius fields with negative second variation.
math.DG↗
Rigidity in the Ginzburg--Landau approximation of harmonic spheres
We prove that not every harmonic map from $S^{2}$ to $S^{2}$ can arise as a limit of Ginzburg--Landau critical points. More precisely, we show that the only degree-one harmonic maps that can be approximated in this way are rotations. This conclusion follows from a rigidity theorem: we show that for every $\gamma>0$ and $\varepsilon$ small enough, the only critical points $u_\varepsilon:S^{2}\to\mathbb R^{3}$ of the Ginzburg--Landau energy $E_\varepsilon$ with energy below $8\pi-\gamma$ are (up to conjugation) rotations, that is $u_\varepsilon(x)=\sqrt{1-2\varepsilon^{2}}\;R_\varepsilon\,x$.
math.DG↗
Morse Index Stability for the Ginzburg-Landau Approximation
In this paper we study the behaviour of critical points of the Ginzburg-Landau perturbation of the Dirichlet energy into the sphere $E_\varepsilon(u):=\int_\Sigma \frac{1}{2}|du|^2_h\ \,dvol_h +\frac{1}{4\varepsilon^2}(1-|u|^2)^2\,dvol_h=\int_{\Sigma}e_{\varepsilon}(u)$. Our first main result is a precise point-wise estimate for $e_\varepsilon(u_k)$ in the regions where compactness fails, which also implies the $L^{2,1}$ quantization in the bubbling process. Our second main result consists in applying the method developed in a previous joint paper with T. Rivi\`ere to study the upper-semi-continuity of the extended Morse index to sequences of critical points of $E_{\epsilon}$: given a sequence of critical points $u_{\varepsilon_k}:\Sigma\to \mathbb{R}^{n+1}$ of $E_\varepsilon$ that converges in the bubble tree sense to a harmonic map $u_\infty\in W^{1,2}(\Sigma,{S}^{n})$ and bubbles $v^i_{\infty}:\mathbb{R}^2\to {S}^{n}$, we show that the extended Morse indices of the maps $v^i,u_\infty$ control the extended Morse index of the sequence $u_{\varepsilon_k}$ for $k$ large enough.
math.DG↗
Optimal weighted Wente's inequality
We prove $L^\infty$ and $W^{1,2}$ weighted Wente's inequalities. We prove in particular the critical case: for the $|x|^2$ weighted Wente's estimate the optimal weight is $|x|^2\log|x|$.
math.AP↗
Morse Index Stability for Critical Points to Conformally invariant Lagrangians
We prove the upper-semi-continuity of the Morse index plus nullity of critical points to general conformally invariant Lagrangians in dimension 2 under weak convergence. Precisely we establish that the sum of the Morse indices and the nullity of an arbitrary sequence of weakly converging critical points to a general conformally invariant Lagrangians of maps from an arbitrary closed surface into an arbitrary closed smooth manifold passes to the limit in the following sense : it is asymptotically bounded from above by the sum of the Morse indices plus the nullity of the weak limit and the bubbles, while it was well known that the sum of the Morse index of the weak limit with the Morse indices of the bubbles is asymptotically bounded from above by the Morse indices of the weakly converging sequence. The main result is then extended to the case of sequences of maps from sequences of domains degenerating to a punctured Riemann surface assuming that the lengths of the images by the maps of the collars associated to this degeneration stay below some critical length.
math.DG↗