On Liouville Theorems For Vector-Bundle-Valued Exponential Forms

In this paper, we introduce exponential energy functional, exponential stress-energy tensor and exponential conservation law for vector bundle valued forms, in or-der to obtain exponential monotonicity inequalities and vanishing theorems through the coarea formula and comparison theorems in Riemannian geometry. Meanwhile, we introduce the exponential Yang-Mills functional and its variational formula, and also prove that the exponential conservation law holds for exponential Yang-Mills field. In particular, we obtain vanishing theorems for exponentially harmonic map and exponen-tial Yang-Mills field, since they both satisfy the exponential conservation law.