# On the heat flow on metric measure spaces: existence, uniqueness and stability

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We locally classify the 3-dimensional generalized ($\kappa,\mu$)-contact metric manifolds, which satisfy the condition $\Vert grad\kappa \Vert =$const. ($\not=0$). This class of manifolds is determined by two arbitrary functions of one variable.

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Let $$(M,g)$$ be a two dimensional compact Riemannian manifold of genus $$g(M)>1$$ . Let $$f$$ be a smooth function on $$M$$ such that Let $$p_1,\ldots ,p_n$$ be any set of points at which $$f(p_i)=0$$ and $$D^2f(p_i)$$ is non-singular. We prove that for all sufficiently small $$\lambda >0$$...