论文标题
在不完整的riemannian流形上，对热核的对数衍生物的局部界限
Localized bounds on log-derivatives of the heat kernel on incomplete Riemannian manifolds
论文作者
论文摘要
紧凑的riemannian歧管上热核的对数衍生物的界限已知已知，并且最近被扩展到了对数学者和婚姻中的范围，以使一般完整的Riemannian歧管。在这里，我们将这些界限进一步扩展到在无限范围内的最小限制条件下，对于所有订单的衍生物，在最小的限制条件下。此外，我们不仅考虑了与Laplace-Beltrami操作员相关的通常的热核，而且还允许增加保守的向量领域。我们表明，即使对于紧凑的歧管，这些界限通常都很尖锐，并且我们讨论了当操作员合并非保守矢量场或riemannian结构削弱到亚利曼式结构时会出现的困难。
Bounds on the logarithmic derivatives of the heat kernel on a compact Riemannian manifolds have been long known, and were recently extended, for the log-gradient and log-Hessian, to general complete Riemannian manifolds. Here, we further extend these bounds to incomplete Riemannan manifolds under the least restrictive condition on the distance to infinity available, for derivatives of all orders. Moreover, we consider not only the usual heat kernel associated to the Laplace-Beltrami operator, but we also allow the addition of a conservative vector field. We show that these bounds are sharp in general, even for compact manifolds, and we discuss the difficulties that arise when the operator incorporates non-conservative vector fields or when the Riemannian structure is weakened to a sub-Riemannian structure.
