The curl operator and Sobolev inequalities for differential forms
Dienstag, 28.10.25, 16:15-17:45, Seminarraum 125
The curl operator for vectors in R^3 is of special importance and gives rise to various Sobolev inequalities. In this talk we will introduce the generalized curl operator for differential forms in higher dimensions and discuss the spectral analysis. As an application, we prove that fundamental bubbles (Killing forms) are local minimizers of one Sobolev inequality, but not local minimizers of another Sobolev inequality. This is a joint work with Prof. Guofang Wang.
Anisotropic minimal graphs with free boundary
Dienstag, 4.11.25, 16:15-17:45, Seminarraum 125
Minimal surface equation is a classical topic in Geometric Analysis and PDEs. In this talk, we discuss recent progress on anisotropic minimal surface equation, and prove the following Liouville-type theorem: any anisotropic minimal graph with free boundary in the half-space must be flat, provided that the graph function has at most one-side linear growth. This is a joint work with Guofang Wang, Wei Wei, and Chao Xia.
The stability of Sobolev inequality on the Heisenberg group
Dienstag, 11.11.25, 16:15-17:45, Seminarraum 125
In this talk, we are concerned with the optimal asymptotic lower bound for the stability of Sobolev inequality on the Heisenberg group. We first establish the optimal local stability of Sobolev inequality on the CR sphere through bispherical harmonics and complicated orthogonality technique (see Lemma 3.1). The loss of the Polya-Szego inequality and the Riesz rearrangement inequality on the Heisenberg group makes it impossible to use any rearrangement flow technique to derive the optimal stability of Sobolev inequality on the CR sphere from corresponding optimal local stability. To circumvent this, we will use the CR Yamabe flow to pass from the local stability to the global stability and thus establish the optimal stability of Sobolev inequality on the Heisenberg group with the dimension-dependent constants (see Theorem 1.1). This work was accomplished together with Lu Chen, Guozhen Lu and Hanli Tang.
Stability of the Clifford Torus as a Willmore Minimizer
Dienstag, 18.11.25, 16:15-17:45, Seminarraum 125
This is joint work with Jie Zhou (Capital Normal University). We prove that surfaces in \(\mathbb{S}^3\) with genus \(\geq 1\) and Willmore energy \(\leq 2\pi^2 + \delta^2\) are quantitatively close to the Clifford torus after a conformal transformation. The closeness is measured in three aspects: \(W^{2,2}\) parametrization, \(L^\infty\) conformal factor, and conformal structure, with linear dependence on \(\delta\).
The super-Liouville equation on the sphere
Dienstag, 2.12.25, 16:15-17:45, Seminarraum 125
In this work, we study the super-Liouville equation on the sphere with positive coefficient functions. We begin by deriving estimates for the spinor component of the equation, thereby finding that the energy of the spinor part of solutions is uniformly bounded. We then analyze the compactness of the solution space in two aspects: the compactness for solutions with small energy and the compactness with respect to the conformal transformation group of the sphere. Finally, by introducing a new natural constraint, the Nehari manifold, and employing variational methods, we obtain the existence of the least energy solutions when the coefficient functions are even.