Quantization for an evolution equation with critical exponential growth on a closed Riemann surface

In this paper, we analyze the concentration behavior of a positive solution to an evolution equation with critical exponential growth on a closed Riemann surface, and particularly derive an energy identity for such a solution. This extends a result of Lamm-Robert-Struwe and complements that of Yang.     

A heat flow for the critical Trudinger-Moser functional on a closed Riemann surface

Annals of Global Analysis and Geometry - In this paper, we propose a heat flow for the critical Trudinger-Moser functional on a closed Riemann surface $$(Sigma ,g)$$ . We prove its short time...     

Non-existence of extremals for the Adimurthi–Druet inequality

The Adimurthi–Druet [1] inequality is an improvement of the standard Moser–Trudinger inequality by adding a $L^2$-type perturbation, quantified by $\alpha \in [0,{\lambda }_1)$, where ${\lambda }_1$ is the first Dirichlet eigenvalue of Δ on a smooth bounded domain. It is known [3], [10], [14], [19] that this inequality admits extremal functions, when the perturbation parameter α is small. By contrast, we prove here that the Adimurthi–Druet inequality does not admit any extremal, when the perturbation parameter α approaches ${\lambda }_1$. Our result is based on sharp expansions of the Dirichlet energy for blowing sequences of solutions of the corresponding Euler–Lagrange equation, which take into account the fact that the problem becomes singular as $\alpha \to {\lambda }_1$.     

The Riemann problem on a finite-sheeted Riemann surface of infinite genus

LetR be the Riemann surface of the functionu(z) specified by the equationu ⁿ=P(z) withn ε ℕ,n ≥ 2, andz ε ℂ, whereP(z) is an entire function with infinitely many simple zeros. OnR, the Riemann boundary-value problem for an arbitrary piecewise smooth contour Γ is considered. Necessary and sufficient conditions for its solvability are obtained, and its explicit solution is constructed. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 25–35, January, 2000.     

Quasiconformal extremals of smooth functionals and of the energy integral on Riemann surfaces

Krasnodar. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 3, pp. 163–174, May–June, 1988.     

Coupled connections on a compact Riemann surface

We consider holomorphic differential operators on a compact Riemann surface X whose symbol is an isomorphism. Such a differential operator of order n on a vector bundle E sends E to K^⊗n_X⊗E, where K_X is the holomorphic cotangent bundle. We classify all those holomorphic vector bundles E over X that admit such a differential operator. The space of all differential operators whose symbol is an isomorphism is in bijective correspondence with the collection of pairs consisting of a flat vector bundle E over X and a holomorphic subbundle of E satisfying a transversality condition with respect to the connection.     

Remark on the size of a Riemann surface

Research partially supported by the National Science Foundation.     

The isoperimetric inequality on a surface

We prove a new isoperimetric inequality which relates the area of a multiply connected curved surface, its Euler characteristic, the length of its boundary, and its Gaussian curvature. Received: 31 July 1998