arXiv research

Proves convergence of gradient Ricci shrinkers with uniform bounds.

Study proves Lojasiewicz inequalities for harmonic maps near simple bubble trees.

Maps converge to simpler structures under certain tension conditions.

Study on prescribing positive curvature with conical singularities on a sphere.

The study proves compactness of Hamiltonian stationary Lagrangian surfaces in Kähler surfaces.

In this paper, we study the weak compactness of the set of conformal metrics in any Riemann surface without boundary whose Calabi energy and area are uniformly bounded. We prove that for any sequence of such metrics, there alwasy exists a subsequence which converges in H\sp{2,2}_\sb{loc} everywhere except a finite numb…

Compactness theorem for quasiregular curves proves normality and resolves nodal points.

Paper proves Łojasiewicz inequalities near simple bubble trees on surfaces.

We determine bubble tree convergence for a sequence of harmonic maps, with uniform energy bounds, from a compact Riemann surface into a compact locally CAT(1) space. In particular, we demonstrate energy quantization and the no-neck property for such a sequence. In the smooth setting, Jost and Parker respectively establ…

Proves finite step termination of Kähler-Einstein metric singularity formation.

We construct a partial order relation which acts on the set of 3-cliques of a maximal planar graph G and defines a unique hierarchy. We demonstrate that G is the union of a set of special subgraphs, named `bubbles', that are themselves maximal planar graphs. The graph G is retrieved by connecting these bubbles in a tre…

The paper proves a Lojasiewicz inequality for maps from the 2-sphere to itself.

Study on metric bubbles in complex dimensions 1 and 2.

In this paper we develop a bubble tree structure for a degenerating class of Riemannian metrics satisfying some global conformal bounds on compact manifolds of dimension 4. Applying the bubble tree structure, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and a diameter bound f…

Let $(M; g)$ be a smooth compact Riemiannian manifold without boundary and $g_{k}$ be a metric conformal to $g$. Suppose $vol(M; g_{k})+||R_{k}||_{L^{p}(M;g_{k})} < C$, where $R_{k}$ is the scalar curvature and $p > \frac{n}{2}$. We will use the 3-circle theorem and the John-Nirenberg inequality to study the bubble tre…

We explore geometric aspects of bubble convergence for harmonic maps. More precisely, we show that the formation of bubbles is characterised by the local excess of curvature on the target manifold. We give a universal estimate for curvature concentration masses at each bubble point and show that there is no curvature l…

Gluing theorem for collapsing warped-QAC Calabi-Yau manifolds verified.

The paper develops a theory for free boundary minimal surfaces with genus at least one.

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 which allows us to find area constrained, minimizing, immersed, haunted bubble …

We study a class of weakly conformal $3$-harmonic maps, called associative Smith maps, from $3$-manifolds into $7$-manifolds that parametrize associative $3$-folds in Riemannian $7$-manifolds equipped with $\mathrm{G}_2$-structures. Associative Smith maps are solutions of a conformally invariant nonlinear first order P…

The paper proves a new method to find the maximum Laplace eigenvalues on surfaces.

Study of critical points in Ginzburg-Landau approximation with stability results.

The goal of the present work is twofold. First we prove the existence of an Hilbert Manifold structure on the space of immersed oriented closed surfaces with three derivatives in $L^2$ in an arbitrary sub-manifold $M^m$ of an euclidian space $R^Q$. Second, using this Hilbert manifold structure, we prove a lower semi co…

Study on membranes under confinement, proving existence and regularity of minimizers.

The paper studies vector bundles over surfaces, focusing on singularity formation.

This paper proves compactness of conformal Chern-minimal surfaces in Hermitian surfaces.

Study on CMC hypersurfaces with bounded index and area, proving multiplicity one convergence and bounds on genus.

Two-cycle GEILA equilibria are OLG equilibria and vice versa, with applications to indeterminacy and bubbles.

We present some geometric applications, of global character, of the bubbling analysis developed by Buzano and Sharp for closed minimal surfaces, obtaining smooth multiplicity one convergence results under upper bounds on the Morse index and suitable lower bounds on either the genus or the area. For instance, we show th…

For the class of approximate harmonic maps $u\in W^{1,2}(Σ,N)$ from a closed Riemmanian surface $(Σ,g)$ to a compact Riemannian manifold $(N, h)$, we show that (i) the so-called energy identity holds for weakly convergent approximate harmonic maps $\{u_n\}:Σ\to N$, with tension fields $τ(u_n)$ bounded in the Morrey spa…

In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus $g\geq 2$. We develop a direct variational methods similar to the proof of the famous Plateau problem by J. Douglas and T. Rado. As a result, we show that the min-max value for the area functional can be achieved by a …

We discuss - in what is intended to be a pedagogical fashion - a criterion, which is a lower bound on a certain ratio, for when a stock (or a similar instrument) is not a good investment in the long term, which can happen even if the expected return is positive. The root cause is that prices are positive and have skewe…

Study soap bubbles with almost constant higher-order mean curvature, proving unique limits under certain conditions.

We define a family of functionals generalizing the Yang-Mills functional. We study the corresponding gradient flows and prove long-time existence and convergence results for subcritical dimensions as well as a bubbling criterion for the critical dimensions. Consequently, we have an alternate proof of the convergence of…

Small bubbles sliding on a boundary maintain half-spherical shape.

We develop a bubble tree construction and prove compactness results for $W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in a compact Riemannian manifold with uniformly bounded areas and Willmore energies. The compactness property is appli…

Wave maps can have multiple bubbling solutions at blow-up points.

In this paper we prove a convergence result for sequences of Willmore immersions with simple minimal bubbles. To this end we replace the total curvature control in T. Rivière's proof of the $\varepsilon$-regularity for Willmore immersions by a control of the local Willmore energy.

Profinite rigidity proven for many hyperbolic manifolds.

We quantify the amount of information filtered by different hierarchical clustering methods on correlations between stock returns comparing it with the underlying industrial activity structure. Specifically, we apply, for the first time to financial data, a novel hierarchical clustering approach, the Directed Bubble Hi…

This paper addresses the statistical properties of time series driven by rational bubbles a la Blanchard and Watson (1982), corresponding to multiplicative maps, whose study has recently be revived recently in physics as a mechanism of intermittent dynamics generating power law distributions. Using insights on the beha…

Given a principal bundle $P\to M$ over a Riemannian manifold with compact structure group $G$, let us consider a stationary Yang-Mills connection $A$ with energy $\int_M |F_A|^2\le Λ$. If we consider a sequence of such connections $A_i$, then it is understood that up to subsequence we can converge $A_i\to A$ to a singu…

We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a general quantization identity for the total curvature functional, valid in ambient di…

Stability of Yang-Mills connections' Morse indices and nullity in 4D.

Study proves upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces.

Preserves scalar curvature bounds under weak convergence of 3-manifolds.

Study on Yang-Mills heat flow on $\mathbb{R}^4$ bundles, showing infinite time bubbling.

We prove existence and regularity of minimisers for the Canham-Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the $2$-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to prove the main res…