Poincar\'e duality, Hilbert complexes and geometric applications

Let $(M,g)$ an open and oriented riemannian manifold. The aim of this paper is to study some properties of the two following sequences of $L^2$ cohomology groups: $H^i_{2,m\rightarrow M}(M,g)$ defined as the image \im(H^i_{2,min}(M,g)\rightarrow H^i_{2,max}(M,g)) and $\bar{H}^i_{2,m\rightarrow M}(M,g)$ defined as \im(\bar{H}^i_{2,min}(M,g)\rightarrow \bar{H}^i_{2,max}(M,g)). We show, under certain hypothesis, that the first sequence is the cohomology of a suitable Hilbert complex which contains the minimal one and is contained in the maximal one. We also show that when the second sequence is finite dimensional then Poincar\'e duality holds for it and that, in the same assumptions, when $dim(M)\equiv0\ mod\ 4$ we can use it to define a $L^2$ signature on $M$. Moreover we show several applications to the intersection cohomology of compact smoothly stratified pseudomanifolds and we get some results about the Friedrichs extension $\Delta_{i}^\mathcal{F}$ of $\Delta_{i}$.Comment: Final version. To appear on Advances in Mathematics. Comments are welcom

On the L2−L^2-Poincar\'e duality for incomplete riemannian manifolds: a general construction with applications

Let $(M,g)$ be an open, oriented and incomplete riemannian manifold of dimension $m$. Under some general conditions we show that it is possible to build a Hilbert complex $(L^2\Omega^i(M,g),d_{\mathfrak{M},i})$ such that its cohomology groups, labeled with $H^i_{2,\mathfrak{M}}(M,g)$, satisfy the following properties: \begin{itemize} \item H^i_{2,\mathfrak{M}}(M,g)=ker(d_{max,i})/\im(d_{min,i}) \item $H^i_{2,\mathfrak{M}}(M,g)\cong H^{m-i}_{2,\mathfrak{M}}(M,g)$ (Poincar\'e duality holds) \end{itemize} Finally in the rest of the paper we study some properties of this complex with particular attention to the sufficient conditions which make it a Fredholm complex.Comment: Final version. To appear on Journal of Topology and Analysi

Twisted K-homology,Geometric cycles and T-duality

Twisted $K$-homology corresponds to $D$-branes in string theory. In this paper we compare two different models of geometric twisted $K$-homology and get their equivalence. Moreover, we give another description of geometric twisted $K$-homology using bundle gerbes. We establish some properties of geometric twisted $K$-homology. In the last part we construct $T$-duality isomorphism for geometric twisted $K$-homology.Comment: We modify the statement about the six-term exact sequence of geometric twisted $K$-homology. Some Typos are corrected. Comments are welcome

Sobolev spaces and Bochner Laplacian on complex projective varieties and stratified pseudomanifolds

Let $V\subset \mathbb{C}\mathbb{P}^n$ be an irreducible complex projective variety of complex dimension $v$ and let $g$ be the K\"ahler metric on \reg(V), the regular part of $V$, induced by the Fubini Study metric of $\mathbb{C}\mathbb{P}^n$. In this setting Li and Tian proved that W^{1,2}_0(\reg(V),g)=W^{1,2}(\reg(V),g), that the natural inclusion W^{1,2}(\reg(V),g)\hookrightarrow L^2(\reg(V),g) is a compact operator and that the heat operator associated to the Friedrich extension of the scalar Laplacian \Delta_0:C^{\infty}_c(\reg(V))\rightarrow C^{\infty}_c(\reg(V)), that is e^{-t\Delta_0^{\mathcal{F}}}:L^2(\reg(V),g)\rightarrow L^2(\reg(V),g), is a trace class operator. The goal of this paper is to provide an extension of the above result to the case of Sobolev spaces of sections and symmetric Schr\"odinger type operators with potential bounded from below where the underling riemannian manifold is the regular part of a complex projective variety endowed with the Fubini-Study metric or the regular part of a stratified pseudomanifold endowed an iterated edge metric.Comment: Final version. To appear on The Journal of Geometric Analysi

On the Laplace-Beltrami operator on compact complex spaces

Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $v>1$. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corresponding eigenvalues and we use these estimates to deduce that the associated heat operators are trace-class. Finally we give various applications to the Hodge-Dolbeault operator and to the Hodge-Kodaira Laplacian in the setting of Hermitian complex spaces of complex dimension $2$.Comment: To appear on Transactions of the American Mathematical Society. Comments are welcome. arXiv admin note: text overlap with arXiv:1607.0028

Anisotropic Wetting Property of Superhydrophobic Surfaces and Electrokinetic Flow on Liquid-Filled Surfaces

Understanding the wetting property of rough surface is critical in guiding droplets and novel superhydrophobic surface design. The Cassie-Baxter model and Wenzel model are always used to describe the totally non-wetting and completely wetting states, however, there were few discussions about the intermediate state. Through measuring the contact angles of groove patterned surfaces in different groove orientations, the anisotropic wetting properties of groove patterned superhydrophobic surface were investigated. The degree of water penetration into the grooves was experimentally observed and it was found that the degree of water penetration was different with groove orientations, which would affect the corresponding contact angle. Besides guiding droplets, superhydrophobic surfaces are also very important in microfluidic due to their ability to generate fluid slip and flow enhancement. After a deeper understanding of the wetting property of groove patterned superhydrophobic surface, I further investigated its important role in microfluidics. In this dissertation, I mainly focus on electrokinetics on groove patterned surface and liquid-filled slippery surfaces, a new kind of surface by filling low surface tension oil into the interstices of groove patterned surfaces. I experimentally measured the streaming potential on flat parylene surface, air-filled groove patterned surface and liquid-filled surfaces and compared their effects in streaming potential enhancement. The liquid-filled surfaces were shown to be able to enhance the generated streaming potential due to its slippery property and liquid-oil interface charges. As the electrokinetic on liquid-filled surfaces is a new phenomenon, the underlying physics is still not clear. I further investigated the influences of filled oil properties and groove orientation on streaming potentials and fluid slip. Oils with different densities, viscosities, dielectric constant, conductivities and surface tensions were filled into the interstices of groove patterned surfaces to make different types of liquid-filled surfaces. The streaming potentials on liquid-filled surfaces with different oils were experimentally measured. An empirical relationship between streaming potential and oil properties was found and the effects of electrical properties, such as interface charge density and dielectric constant of filled oil, on fluid slip were also studied. Finally, the groove orientation was varied to study the tensorial effects on streaming potential. Through both streaming potential measurement and theoretical analysis, it was found that the streaming potential at 45° was always smaller than the arithmetic mean of those at 0° and 90°, and the pressure gradient in the transvers direction generated by tensorial effects was important in the streaming potential modification. My work will be important in guiding droplets, flow patterning, lab-on-chip devices and the development of electrokientic based power sources

Degenerating Hermitian metrics and spectral geometry of the canonical bundle

Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $m$. In this paper we are interested in the Dolbeault operator acting on the space of $L^2$ sections of the canonical bundle of $reg(X)$, the regular part of $X$. More precisely let $\overline{\mathfrak{d}}_{m,0}:L^2\Omega^{m,0}(reg(X),h)\rightarrow L^2\Omega^{m,1}(reg(X),h)$ be an arbitrarily fixed closed extension of $\overline{\partial}_{m,0}:L^2\Omega^{m,0}(reg(X),h)\rightarrow L^2\Omega^{m,1}(reg(X),h)$ where the domain of the latter operator is $\Omega_c^{m,0}(reg(X))$. We establish various properties such as closed range of $\overline{\mathfrak{d}}_{m,0}$, compactness of the inclusion $\mathcal{D}(\overline{\mathfrak{d}}_{m,0})\hookrightarrow L^2\Omega^{m,0}(reg(X),h)$ where $\mathcal{D}(\overline{\mathfrak{d}}_{m,0})$, the domain of $\overline{\mathfrak{d}}_{m,0}$, is endowed with the corresponding graph norm, and discreteness of the spectrum of the associated Hodge-Kodaira Laplacian $\overline{\mathfrak{d}}_{m,0}^*\circ \overline{\mathfrak{d}}_{m,0}$ with an estimate for the growth of its eigenvalues. Several corollaries such as trace class property for the heat operator associated to $\overline{\mathfrak{d}}_{m,0}^*\circ \overline{\mathfrak{d}}_{m,0}$, with an estimate for its trace, are derived. Finally in the last part we provide several applications to the Hodge-Kodaira Laplacian in the setting of both compact irreducible Hermitian complex spaces with isolated singularities and complex projective surfaces.Comment: Final version. To appear on Advances in Mathematic