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We study polynomial growth holomorphic functions and forms on complete gradient shrinking Ricci solitons. By relating to the spectral data of the $f$-Laplacian, we show that the dimension of the space of polynomial growth holomorphic functions or holomorphic $(p,0)$-forms are finite. In particular, a sharp dimension estimate for the space of linear growth holomorphic functions was obtained. Under some additional curvature assumption, we prove a sharp estimate for the frequency of polynomial growth holomorphic functions, which was used to obtain dimension upper bound as a power function of the polynomial order.

In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate : any bounded harmonic function is constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension.

In this paper we establish the three balls theorem for functions $u$ satisfying $Du=\lambda u$ in Clifford analysis, where $D$ is the Dirac operator. As an application, we generalize Hadamard's three circles theorem to monogenic function in $\mathbb R^{n+1}.$

We prove rigidity theorems for shrinking gradient Ricci solitons supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in $\mathbb{R}^n$. In addtion, we partially give analogous rigidity results of the Caffarelli-Kohn-Nirenberg inequalities on shrinking Ricci solitons.

An isometric compact group action $G \times (M,g) \rightarrow (M,g)$ is called polar if there exists a closed embedded submanifold $\Sigma \subseteq M$ which meets all orbits orthogonally. Let $\Pi$ be the associated generalized Weyl group. We study the properties of the lifting action $G$ on the cotangent bundle $T^*M$. In particular, we show that the restriction map $(C^{\infty}(T^*M))^G \rightarrow (C^{\infty}(T^* \Sigma))^{\Pi}$ is a surjective homomorphism of Poisson algebras. As a corollary, the singular symplectic reductions $T^*M // G $ and $T^* \Sigma // \Pi$ are isomorphic as stratified symplectic spaces, which gives a partial answer to a conjecture of Lerman, Montgomery and Sjamaar.