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The answer is "Yes" by the following harmonic analogy of Schwarz lemma: See Proposition 1.5. of this paper which says that $4/\pi$ is a sharp upper bound. https://arxiv.org/pdf/1010.4 …

Let $n>1$ and $M\subset \mathbb{R}^n$ be a (sufficiently low dimensional) compact analytic submanifold. Assume that $f:\mathbb{R}^n\to \mathbb{R}$ is an analytic function. Is there a Harmonic funct …

Let $f:\mathbb{R}^{n}\to \mathbb{R}$ be a smooth function. What type of obstructions exist for existence of a Riemannian metric $g$ on $\mathbb{R}^{n}$ such that $f$ is a harmonic functi …

A map or a vector field $g: \mathbb{R}^n \to \mathbb{R}^n $ is called a harmonic map if all its components are harmonic functions. Motivated by conversations on this questions we ask: …

Let $X$ be a non-vanishing real analytic vector field on an open manifold $M$. What kind of obstructions would appear when we search for a Riemannian metric on $M$ such that the space of …

Is there an smooth foliation of the plane which is not smoothly equivalent to a foliation $dH=0$ where H is a harmonic function without critical values? If the answer is negative then we conclude tha …

Edit according to the comment of user36931 I remove the "motivation" from the previous version and I add an statement to the first question We consider the following two classes of smooth maps on …