arXiv research

Paper proves Łojasiewicz inequalities near simple bubble trees on surfaces.

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

Proves convergence of gradient Ricci shrinkers with uniform bounds.

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

Maps converge to simpler structures under certain tension conditions.

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…

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

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 …

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

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

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

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

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

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…

Study on metric bubbles in complex dimensions 1 and 2.

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…

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…

We prove a bubble tree convergence theorem for a sequence of closed Hamiltonian Stationary Lagrangian surfaces with bounded areas and Willmore energies in a complete K{ä}hler surface. We also prove two strong compactness theorems on the space of Hamiltonian stationary Lagrangian tori in $\mathbb C^2$ and $\mathbb{CP}^2…

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…

Harmonic map flow's singularity properties proven with Lojasiewicz inequalities.

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…

Proves Li-Yau inequality for Helfrich functional, ensuring embeddedness in spherical cases.

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 …

This is a survey paper of our current research on the theory of partial differential equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be …

Study eigenvalues and eigenfunctions of fourth-order operators in annuli, proving optimal estimates and non-radiality.

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

We introduce a notion of generalized Willmore functionals motivated by the Hawking energy of General Relativity and bending energies of membranes. An example of a bending energy is discussed in detail. Using results of Y. Chen and J. Li, we present a compactness result for branched, immersed, haunted, stratified surfac…

Closed Riemannian 4 or 5-manifolds contain branched immersed closed minimal surfaces.

The paper proves continuity of Morse index for Ricci shrinkers.

We generalize our previous results (Theorem 1 and Corollary 2 in arXiv:1412.4114) and Theorem 1 in arXiv:1502.00668) on the existence of an $L^2$-energy gap for Yang-Mills connections over closed four-dimensional manifolds and energies near the ground state (occupied by flat, anti-self-dual, or self-dual connections) t…

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…

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

Let $X$ be a closed, four-dimensional, oriented, smooth manifold with a Riemannian metric, $g$, let $G$ be a compact Lie group, and $P$ be a principal $G$ bundle over $X$. D. Groisser and T. Parker (1987, 1989) and S. K. Donaldson (1990) conjectured that the moduli space of $g$-anti-self-dual connections on $P$, endowe…

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…