On compact Hermitian manifolds with flat Gauduchon connections

Given a Hermitian manifold $(M^n,g)$, the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call $\nabla^s = (1-\frac{s}{2})\nabla^c + \frac{s}{2}\nabla^b$ the $s$-Gauduchon connection of $M$, where $\nabla^c$ and $\nabla^b$ are respectively the Chern and Bismut connections. It is natural to ask when a compact Hermitian manifold could admit a flat $s$-Gauduchon connection. This is related to a question asked by Yau \cite{Yau}. The cases with $s=0$ (a flat Chern connection) or $s=2$ (a flat Bismut connection) are classified respectively by Boothby \cite{Boothby} in the 1950s or by Q. Wang and the authors recently \cite{WYZ}. In this article, we observe that if either $s\geq 4+2\sqrt{3} \approx 7.46$ or $s\leq 4-2\sqrt{3}\approx 0.54$ and $s\neq 0$, then $g$ is K\"ahler. We also show that, when $n=2$, $g$ is always K\"ahler unless $s=2$. Note that non-K\"ahler compact Bismut flat surfaces are exactly those isosceles Hopf surfaces by \cite{WYZ}.Comment: 9 pages. This preprint was submitted to Acta Mathematica Sinica, a special issue dedicated to Professor Qikeng L

Proton block of proton-activated TRPV1 current.

The TRPV1 cation channel is a polymodal nociceptor that is activated by heat and ligands such as capsaicin and is highly sensitive to changes in extracellular pH. In the body core, where temperature is usually stable and capsaicin is normally absent, H(+) released in response to ischemia, tissue injury, or inflammation is the best-known endogenous TRPV1 agonist, activating the channel to mediate pain and vasodilation. Paradoxically, removal of H(+) elicits a transient increase in TRPV1 current that is much larger than the initial H(+)-activated current. We found that this prominent OFF response is caused by rapid recovery from H(+) inhibition of the excitatory current carried by H(+)-activated TRPV1 channels. H(+) inhibited current by interfering with ion permeation. The degree of inhibition is voltage and permeant ion dependent, and it can be affected but not eliminated by mutations to acidic residues within or near the ion selectivity filter. The opposing H(+)-mediated gating and permeation effects produce complex current responses under different cellular conditions that are expected to greatly affect the response of nociceptive neurons and other TRPV1-expressing cells

Smooth quasi-developable surfaces bounded by smooth curves

Computing a quasi-developable strip surface bounded by design curves finds wide industrial applications. Existing methods compute discrete surfaces composed of developable lines connecting sampling points on input curves which are not adequate for generating smooth quasi-developable surfaces. We propose the first method which is capable of exploring the full solution space of continuous input curves to compute a smooth quasi-developable ruled surface with as large developability as possible. The resulting surface is exactly bounded by the input smooth curves and is guaranteed to have no self-intersections. The main contribution is a variational approach to compute a continuous mapping of parameters of input curves by minimizing a function evaluating surface developability. Moreover, we also present an algorithm to represent a resulting surface as a B-spline surface when input curves are B-spline curves.Comment: 18 page

On Bismut Flat Manifolds

In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure. In particular, isosceles Hopf surfaces are the only Bismut flat compact non-K\"ahler surfaces, while central Calabi-Eckmann threefolds are the only simply-connected compact Bismut flat threefolds.Comment: In this 3rd version, we add a lemma on Hermitian surfaces with flat Riemannian connection. References are updated and typos correcte

The set of all orthogonal complex structures on the flat 66-tori

In \cite{BSV}, Borisov, Salamon and Viaclovsky constructed non-standard orthogonal complex structures on flat tori $T^{2n}_{\mathbb R}$ for any $n\geq 3$. We will call these examples BSV-tori. In this note, we show that on a flat $6$-torus, all the orthogonal complex structures are either the complex tori or the BSV-tori. This solves the classification problem for compact Hermitian manifolds with flat Riemannian connection in the case of complex dimension three.Comment: 14 page