Study of integral flows on Riemannian manifolds with focus on blow-up profiles and concentration-compactness.
problem Analyzing nonlinear integral flows on Riemannian manifolds with specific focus on blow-up profiles and concentration-compactness.
method Investigation of a family of nonlinear integral flows involving Riesz potentials, focusing on the Hardy-Littlewood-Sobolev (HLS) subcritical and critical regimes.
result Established convergence on unit spheres and certain locally conformally flat manifolds for the dual Yamabe flow.
Proves continuity and singular set dimension for 2D maps with Q values.
problem Interior regularity of 2D Q-valued maps. method Strong concentration-compactness theorem for equicontinuous maps.
result 2D Q-valued maps are Hölder continuous with singular set dimension ≤1. Study convergence of Yamabe flow on singular spaces with positive constant.
problem Analyzing convergence of Yamabe flow on singular spaces.
method Normalized Yamabe flow with positive Yamabe constant on pseudo-manifolds, including stratified spaces.
result Established convergence under low energy condition and investigated alternatives.
We investigate different concentration-compactness phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in R2m, then that of a closed manifold and, finally, the particular case of the sphere S2m. In all cases we allow the sign of the Q-curvature to vary, …
In this paper we study the local regularity of closed surfaces immersed in a Riemannian 3-manifold flowing by Willmore flow. We establish a pair of concentration-compactness alternatives for the flow, giving a lower bound on the maximal time of existence of the flow proportional to the concentration of the curvature an…
The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.
problem Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds.
method Generalized Lions' concentration compactness and rigidity results of Sobolev inequalities on singular spaces.
result Almost extremal functions are close to extremal functions on the round sphere and Euclidean Sobolev inequality.
Solves Yamabe problem on compact manifolds using variational methods.
problem Solving the Yamabe problem on compact Riemannian manifolds.
method Variational approach, conformal transformations, Concentration-Compactness method.
result The Yamabe problem is solvable when the manifold's Yamabe invariant is less than that of the sphere.
In this paper, we consider the generalized lambda constant and the existence of ground states of the generalized Perelman's W-functional from a variational formulation. One result is concerned with the estimation of the generalized λ constant. The other results are about the existence of ground states of generalized …
In this paper, we observe a set of functionals of metrics which are all decrease under the Calabi flow and have uniform lower bound along the flow, which give rise to a set of integral estimates on the curvature flow. Using these estimates, together with weak compactness we obtained in previous papers [8] and [10], we …
Study on extremizers for Sobolev inequality on curved manifolds.
problem Existence of extremizers for the sharp p-Sobolev inequality on Riemannian manifolds with nonnegative curvature. method Nonsmooth concentration compactness methods and Mosco-convergence results for Cheeger energy.
result Almost extremal functions are close to radial Euclidean bubbles and almost zero globally under nonnegative curvature.
In this paper we study the steepest descent L2-gradient flow of the functional $\SW_{λ_1,λ_2}$, which is the the sum of the Willmore energy, λ1-weighted surface area, and λ2-weighted enclosed volume, for surfaces immersed in R3. This coincides with the Helfrich functional with zero `spontaneous curvature'.…
Chen's flow is a fourth-order curvature flow motivated by the spectral decomposition of immersions, a program classically pushed by B.-Y. Chen since the 1970s. In curvature flow terms the flow sits at the critical level of scaling together with the most popular extrinsic fourth-order curvature flow, the Willmore and su…
Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds.
problem Characterizing and proving rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds.
method Analysis of Riemannian manifolds and metric measure spaces with synthetic lower Ricci curvature bounds, using concentration compactness and Polya-Szego inequalities.
result Closed Riemannian manifolds with optimal Sobolev constant are isometric to the sphere, and almost equality implies close measure Gromov-Hausdorff convergence to a spherical suspension.
The paper studies how adding a 'Gauge Mass' term breaks gauge symmetry in Yang-Mills-Higgs systems and analyzes the resulting behavior.
problem Breaking gauge symmetry in Yang-Mills-Higgs systems.
method Analyzing the asymptotic behavior of the system with a 'Gauge Mass' term added.
result The system's behavior is characterized by concentration phenomena and convergence to harmonic maps and minimal energies.