We develop a harmonic-coordinate approach to the identity component $G_\omega(\Sigma_g)$ of the symplectomorphism group of a closed oriented surface of genus $g\ge2$. Using an intrinsic decomposition of the Abel-Jacobi displacement into a global flux part and a zero-average harmonic fluctuation, we introduce the harmonic flux norm and prove its non-degeneracy on the full identity component without Floer theory. We also show the norm is continuous in the $C^0$-topology, deduce that $\mathrm{Ham}(\Sigma_g,\omega)$ is $C^0$-closed inside $G_\omega(\Sigma_g)$, and produce a locally injective harmonic-coordinate chart near the identity. Along the way we derive first-order expansions for the norm, quantitative fixed-point obstructions, and propose a finite-dimensional persistence invariant (the harmonic barcode) associated to the harmonic displacement filtration.
We study one-dimensional Jacobi operators of divergence-gradient type on $\ell^2(\mathbb{Z}_{\geq 0})$, with coefficients generated by the doubling map, $a_n(x)=a(2^n x \mathrm{mod} 1)$. For continuous positive sampling functions, we show that the almost-sure essential spectrum is an interval containing the bottom of t...
Let $X$ carry a transversely holomorphic foliation, equivalently an elliptic involutive structure $V\subset T_{\mathbb C}X$, and let ${\mathcal O}_V$ be its sheaf of leafwise-constant, transversely holomorphic functions. We construct a finite superconnection model for the derived category of coherent ${\mathcal O}_V$-m...
This note supplies genuinely non-fibred examples for the manuscript: Rigidity on the Two-Torus and Sarnak's Conjecture. For every $0<\delta<\tfrac12$, we construct $C^\infty$ Lebesgue-area-preserving pseudo-rotations of $\mathbb{T}^2$ which satisfy the $(C,\delta)$-deviation condition but do not have bounded mean motio...
Let $n\ge3$ and $D_t=\operatorname{diag}(t^{g_1},\ldots,t^{g_n})$ be a positive diagonal dilation family. We study connected embedded Euclidean hypersurfaces whose diagonal images are minimal. The level-set minimality operator splits into coefficients indexed by the pair sums $g_i+g_j$. Under pair-sum nonresonance, min...
We numerically construct the Sasaki--Einstein metric on the link of the cone over the second del Pezzo surface $\mathrm{dP}_2$ and two primitive harmonic basic $(1,1)$-forms at its irregular volume-minimizing Reeb vector. Being toric, the metric in symplectic coordinates is encoded in a single convex function on a poly...
Since the 1980s, the Bogomolov multiplier of a finite group has been known to obstruct rationality in complex algebraic geometry, and more recently it is understood to be responsible for any torsion in the oriented and stable unitary 2-dimensional $G$-equivariant bordism groups $\Omega_2^{SO,G}$ and $\Omega_2^{U,G}$. I...
Eric Samperton, C. Segovia· 0 citations
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