A Summary On The Intellectual Merits And. Broader Impact Of The Project

Before explaining the obtained results in details we start a summary on the intellectual merits and

broader impact of the project.

The PI initiates a new approach (in items 2,5, 6), using the precise large time

asymptotic behavior of solutions of a parabolic equation to study the geometric property of K

manifolds, and to solve the Poincar Lelong equation. The method is effective in proving

sharp and optimal result. The method reminisces the celebrated ergodic theorem of Birkhoff

which connects the space average of a continuous function on the phase

space of a Hamiltonian system with its time average taking along the trajectory (see the second part for

detailed descriptions). This connection is also in some way related to other …show more content…

In item

11 comparison result for viscosity solutions of some first and second order PDEs are proved. This

immediately yields the celebrated Levy-Gromov isoperimetric inequality and its generalization

as consequences. In item 14, a classification result on four dimensional gradient shrinking

solitons with nonnegative isotropic curvature was proved. This result generalizes the earlier result of

Naber, which proves a classification under the stronger assumption of bounded nonnegative

curvature operator.

The research conducted in items 2, 3, 4, 5, 6 are related to Birkhoff ergodic

theorem (which was applied by H. Weyl to understand the retreats and advances of glaciers. Further

understanding of this connection shall be sensational to the subject of partial differential equation and

dynamic system. The research in items 9, 12, 13 are related to the concept of entropy in

thermodynamics, which have impacts to other sciences beyond mathematics. The work in item 1

contributes an advancement in the high energy physics.

In promoting teaching, training, and learning, the PI

advised (including some current students) nine Ph.D students, including two female graduate

students, and served/serves as the faculty mentor for several postdoc visitors, including one SEW

assistant professor at UC San Diego. At UC San Diego the PI teaches the courses