Some Results on the Weighted Yamabe Problem with or Without Boundary

@article{Ho2022SomeRO,
  title={Some Results on the Weighted Yamabe Problem with or Without Boundary},
  author={Pak Tung Ho and Jin‐Hyuk Shin and Zetian Yan},
  journal={SSRN Electronic Journal},
  year={2022}
}
Let $(M^n,g,e^{-\phi}dV_g,e^{-\phi}dA_g,m)$ be a compact smooth metric measure space with boundary with $n\geqslant 3$. In this article, we consider several Yamabe-type problems on a compact smooth metric measure space with or without boundary: uniqueness problem on the weighted Yamabe problem with boundary, characterization of the weighted Yamabe solitons with boundary and the existence of positive minimizers in the weighted Escobar quotient. 

References

SHOWING 1-10 OF 21 REFERENCES

A Yamabe-type problem on smooth metric measure spaces

We describe and partially solve a natural Yamabe-type problem on smooth metric measure spaces which interpolates between the Yamabe problem and the problem of finding minimizers for Perelman's

On the existence of extremals for the weighted Yamabe problem on compact manifolds

The Yamabe problem on manifolds with boundary

A natural question in differential geometry is whether a given compact Riemannian manifold with boundary is necessarily conformally equivalent to one of constant scalar curvature, where the boundary

A generalization of Escobar-Riemann mapping type problem on smooth metric measure spaces.

In this article, we introduce an analogous problem to Yamabe type problem considered by Case, J., which generalizes the Escobar-Riemann mapping problem for smooth metric measure spaces with boundary.

The weighted σk-curvature of a smooth metric measure space

We propose a definition of the weighted $\sigma_k$-curvature of a smooth metric measure space and justify it in two ways. First, we show that the weighted $\sigma_k$-curvature prescription problem is

Vanishing theorems for f-harmonic forms on smooth metric measure spaces☆

Sharp metric obstructions for quasi-Einstein metrics

Conformal invariants measuring the best constants for Gagliardo–Nirenberg–Sobolev inequalities

We introduce a family of conformal invariants associated to a smooth metric measure space which generalize the relationship between the Yamabe constant and the best constant for the Sobolev

Conformal deformation of a Riemannian metric to constant scalar curvature

A well-known open question in differential geometry is the question of whether a given compact Riemannian manifold is necessarily conformally equivalent to one of constant scalar curvature. This