Perelman's functionals on manifolds with non-isolated conical singularities

In this article, we define Perelman's functionals on manifolds with non-isolated conical singularities by starting from a spectral point of view for the Perelman's $\lambda$-functional. (Our definition of non-isolated conical singularities includes isolated conical singularities.) We prove that the spectrum of Schr\"odinger operator $-4\Delta + R$ on manifolds with non-isolated conical singularities consists of discrete eigenvalues with finite multiplicities, provided that scalar curvatures of cross sections of cones have a certain lower bound. This enables us to define the $\lambda$-functional on these singular manifolds, and further, to prove that the infimum of $W$-functional is finite, with the help of some weighted Sobolev inequalities. Furthermore, we obtain some asymptotic behavior of eigenfunctions and the minimizer of the $W$-functional near the singularity, and a more refined optimal partial asymptotic expansion for eigenfunctions near isolated conical singularities. We also study the spectrum of $-4\Delta + R$ and Perelman's functionals on manifolds with more general singularities, i.e. the $r^{\alpha}$-horn singularities which serve as prototypes of algebraic singularities.Comment: Sections are re-ordered, some proofs are simplified, some typos and errors are corrected, revised version, 58 page

Positive mass theorem for asymptotically flat spin manifolds with isolated conical singularities

There has been a lot of interests in Positive Mass Theorems for singular metrics on smooth manifolds. We prove a positive mass theorem for asymptotically flat (AF) spin manifolds with isolated conical singularities or more generally horn singularities. In particular, we allow topological singularities in the space as we do not require the cross sections of the conical singularity to be spherical. Note that the negative mass Schwarzschild metric is AF with a horn singularity.Comment: References added.28 page

Positive mass theorem for asymptotically flat manifolds with isolated conical singularities

We prove the positive mass theorem for asymptotical flat (AF for short) manifolds with finitely many isolated conical singularities. We do not impose the spin condition. Instead we use the conformal blow up technique which dates back to Schoen's final resolution of the Yamabe conjecture.Comment: 23 pages. arXiv admin note: text overlap with arXiv:2310.1328

Locating the source field lines of Jovian decametric radio emissions

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/154647/1/epp320131.pd

Immune microenvironment of cholangiocarcinoma: Biological concepts and treatment strategies

Cholangiocarcinoma is characterized by a poor prognosis with limited treatment and management options. Chemotherapy using gemcitabine with cisplatin is the only available first-line therapy for patients with advanced cholangiocarcinoma, although it offers only palliation and yields a median survival of < 1 year. Recently there has been a resurgence of immunotherapy studies focusing on the ability of immunotherapy to inhibit cancer growth by impacting the tumor microenvironment. Based on the TOPAZ-1 trial, the US Food and Drug Administration has approved the combination of durvalumab and gemcitabine with cisplatin as the first-line treatment of cholangiocarcinoma. However, immunotherapy, like immune checkpoint blockade, is less effective in cholangiocarcinoma than in other types of cancer. Although several factors such as the exuberant desmoplastic reaction are responsible for cholangiocarcinoma treatment resistance, existing literature on cholangiocarcinoma cites the inflammatory and immunosuppressive environment as the most common factor. However, mechanisms activating the immunosuppressive tumor microenvironment contributing to cholangiocarcinoma drug resistance are complicated. Therefore, gaining insight into the interplay between immune cells and cholangiocarcinoma cells, as well as the natural development and evolution of the immune tumor microenvironment, would provide targets for therapeutic intervention and improve therapeutic efficacy by developing multimodal and multiagent immunotherapeutic approaches of cholangiocarcinoma to overcome the immunosuppressive tumor microenvironment. In this review, we discuss the role of the inflammatory microenvironment-cholangiocarcinoma crosstalk and reinforce the importance of inflammatory cells in the tumor microenvironment, thereby highlighting the explanatory and therapeutic shortcomings of immunotherapy monotherapy and proposing potentially promising combinational immunotherapeutic strategies