I asked this on Math.SE and got no answer, so I’ll try my luck here. Let G be a semisimple real Lie group, U(g) its universal enveloping algebra, let Ω be the Casimir element in U(g) and let f be a smooth (or analytic) real-valued function on G. We then have the following notions 1) … Read more
What does it mean, from the geometrical point of view, use (in General Relativity) of the constraints on the metric tensor’s coefficients such that $\Delta g_{ij}=0$? (where $\Delta$ is the Beltrami-Laplace Operator, $g_{ij}$ the metric tensor). With $\Delta g_{ij}=0$, I mean the Laplace-Beltrami operator, applied componentwise to the components of the metric tensor. Thank you … Read more
Let Dn be the closed unit ball in Rn. Let f:Dn→Rn be a real-analytic orientation preserving immersion, and let ω:Dn→Rn be the harmonic map corresponding to the Dirichlet problem imposed by f, i.e. ω|∂Dn=f|∂Dn Does rank(dω)≥n−1 everywhere on Dn? A perhaps easier question: Suppose that dωp is singular, and that p∈Dn lies in the closure … Read more
A harmonic function on a graph is a function on its vertices such that the value at every vertex is the average of the values at its neighbors. Consider the following method for approximating a harmonic function on a graph, given some initial values on each vertex: at each step, pick a random vertex, and … Read more
Do there exist Jordan analytic curves J in the complex plane C, other than circles, with the following property: There exists a harmonic function u in the unbounded component of C∖J, u(z)=0 and |gradu(z)|=1 for z∈J. If we impose mild restrictions on u near ∞, for example that u is bounded from below or from … Read more
Let u be a harmonic function inside the unit ball B(0,1) in R2 so that |u|⩽. Does a function u which satisfies |\nabla u(0)|>1 exist? If not, please prove that |\nabla u(0)|\leqslant1. Thanks. Answer The answer is “Yes” by the following harmonic analogy of Schwarz lemma: See Proposition 1.5. of this paper which says that … Read more
Closed. This question is off-topic. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for MathOverflow. Closed 5 years ago. Improve this question My understanding is that harmonic analysis on the circle (S1) leads to Fourier Series/Integrals whereas harmonic analysis on the sphere (S2) leads to Spherical … Read more
Let $u$ be a real-valued harmonic function on $\mathbb{D}$, which extends continuously to the boundary. I wonder how to prove the inequality $$\int_\mathbb{D} e^{2u} dxdy\leq\dfrac{1}{4\pi}\left(\int_{\partial\mathbb{D}}e^uds\right)^2.$$ There are a lot of properties with harmonic function $u$ (like maximum principle and mean value equality), but I did not know how to handle with this $e^u$. Answer This … Read more
I have a question regarding harmonic maps from all of ${\Bbb R}^2$ into a domain in ${\Bbb R}^2$. Before stating my question in full generality, let me ask a special case of the question first. Is it possible to find two non-constant harmonic functions $u$ and $v$ on ${\Bbb R}^2$ such that $u>v^3$ at every … Read more
By the uniformization theorem, for every genus-0 closed surface M⊂R3, there is a conformal map f:M→S2. Furthermore consider the Dirichlet Energy E(f)=∫M|∇f|2dλM. A critical point of this energy functional is called harmonic map. Now the intersting statement is: For a genus-0 closed surface M, the conformal maps f:M→S2 are equivalent to the harmonic maps. Can … Read more
