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Is there an elliptic operator $D$ on $C^{\infty}(S^2)$ whose principal symbol is not identical to thats of Laplacian but it satisfies $\int_{S^2} fDf =\int_{S^2} f\Delta (f)$ for all $f\in C^{\infty}(S …

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 …

For a closed manifold the laplacian is almost surjective operator since the index of $\Delta$ is zero and there is no a non constant harmonic function. …

We denote by $\Delta$, the Laplacian associated to this Riemannian structure. Are the following two conditions equivalent? …

Assume that $M$ is a compact Riemannian manifold whose Laplacian is denoted by $\Delta$. Assume that the Euler characteristic of $M$ is zero. …

If this correspondence is denoted by $i$ then the Laplacian of a vector field $X$ is defined as $\Delta X=i^{-1} \Delta (i(X))$. … Where the latter Laplacian is the natural Laplacian on the space of differential forms. …

EDIT: According to some comments on this post I revise the title to remove the misunderestanding. Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated t …

Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $. Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ …