Abstract Let G=(V, E) G=(V, E) be a finite connected graph, and let κ: V → R κ: V→ R be a
function such that ∫ _V κ d μ< 0∫ V κ d μ< 0. We consider the following Kazdan–Warner …

Given a smooth function K (x) satisfying a polynomially cone condition and x·∇ K≤ 0, we
prove that there is no solution u∈ C∞(R 2) of the equation-Δ u= K (x) e 2 u on R 2 with u≤ …

Consider a compact Riemannian surface (M, g) with a nonempty boundary and negative
Euler characteristic. Given two smooth non-constant functions f in M and h in∂ M with max …

Answering a question by M. Struwe [26] related to the blow-up behavior in the Nirenberg
problem, we show that the prescribed Q-curvature equation Δ 2 u=(1−| x| p) e 4 u in R 4 …

The current article aims to study the existence of the solutions for the mean field Eq.(1.1) on
a closed Riemann surface (M, g). To serve this, one constructs a non-local gradient-like flow …

These notes originate from a Nachdiplomvorlesung held by the author at ETH during the
spring term of 2021. The main focus of the course was the classical problem initiated by …

By a classical result of Kazdan–Warner, for any smooth sign-changing function f with
negative mean on the torus (M, gb) there exists a conformal metric g= e2ugb with Gauss …

Initiated by the work of Uhlenbeck in late 1970s, we study existence, multiplicity and
asymptotic behavior for minimal immersions of a closed surface in some hyperbolic three …

On flat torus with standard metric gb, two metrics g1 and g2 with unit volume in the conformal
class of gb are said to be equivalent if they have the same Gaussian curvature, denoted by …

In this paper we study the problem, posed by Troyanov (Trans AMS 324: 793–821, 1991), of
prescribing the Gaussian curvature under a conformal change of the metric on surfaces with …