Volume 2023, Issue 803

We prove a conjecture by Ene, Herzog, and Hibi (2011) that the Betti numbers of the binomial edge ideal J G {J_{G}} of a closed graph G coincide with the Betti numbers of its lex initial ideal M G {M_{G}} . We describe the Betti numbers of the ideal M G {M_{G}} .

We obtain new lower bounds for the first non-zero eigenvalue of the scalar sub-Laplacian for 3-Sasaki metrics, improving the Lichnerowicz–Obata-type estimates by Ivanov, Petkov and Vassilev (2013, 2014). The limiting eigenspace is fully described in terms of the automorphism algebra. Our results can be thought of as an analogue of the Lichnerowicz–Matsushima estimate for Kähler–Einstein metrics. In dimension 7, if the automorphism algebra is non-vanishing, we also compute the second eigenvalue for the sub-Laplacian and construct explicit eigenfunctions. In addition, for all metrics in the canonical variation of the 3-Sasaki metric we give a lower bound for the spectrum of the Riemannian Laplace operator, depending only on scalar curvature and dimension. We also strengthen a result pertaining to the growth rate of harmonic functions, due to Conlon, Hein and Sun (2013, 2017), in the case of hyperkähler cones. In this setup we also describe the space of holomorphic functions.

We study the Yamabe flow starting from an asymptotically flat manifold ( M n , g 0 ) (M^{n},g_{0}) . We show that the flow converges to an asymptotically flat, scalar flat metric in a weighted global sense if Y ⁢ ( M , [ g 0 ] ) > 0 Y(M,[g_{0}])>0 , and show that the flow does not converge otherwise. If the scalar curvature is nonnegative and integrable, then the ADM mass at time infinity drops by the limit of the total scalar curvature along the flow.

A (compact) manifold with fibered 𝑃-singularities is a (possibly) singular pseudomanifold M Σ M_{\Sigma} with two strata: an open nonsingular stratum M ̊ \mathring{M} (a smooth open manifold) and a closed stratum β ⁢ M \beta M (a closed manifold of positive codimension), such that a tubular neighborhood of β ⁢ M \beta M is a fiber bundle with fibers each looking like the cone on a fixed closed manifold 𝑃. We discuss what it means for such an M Σ M_{\Sigma} with fibered 𝑃-singularities to admit an appropriate Riemannian metric of positive scalar curvature, and we give necessary and sufficient conditions (the necessary conditions based on suitable versions of index theory, the sufficient conditions based on surgery methods and homotopy theory) for this to happen when the singularity type 𝑃 is either Z / k \mathbb{Z}/k or S 1 S^{1} , and 𝑀 and the boundary of the tubular neighborhood of the singular stratum are simply connected and carry spin structures. Along the way, we prove some results of perhaps independent interest, concerning metrics on spin 𝑐 manifolds with positive “twisted scalar curvature,” where the twisting comes from the curvature of the spin 𝑐 line bundle.

For a simple, simply connected complex affine algebraic group 𝐺, we prove the existence of a flat projective connection on the bundle of nonabelian theta functions on the moduli spaces of semistable parabolic 𝐺-bundles for families of smooth projective curves with marked points.

We prove that, for a compact, 3-dimensional, real-analytic, Lorentzian manifold, if the group of conformal transformations does not preserve any metric in the conformal class, then the metric is conformally flat.

We show that convex viscosity solutions of the Lagrangian mean curvature equation are regular if the Lagrangian phase has Hölder continuous second derivatives.

In this paper, we investigate the connection between the index and the geometry and topology of capillary surfaces. We prove an index estimate for compact capillary surfaces immersed in general 3-manifolds with boundary. We also study noncompact capillary surfaces with finite index and show that, under suitable curvature assumptions, such surface is conformally equivalent to a compact Riemann surface with boundary, punctured at finitely many points. We then prove that a weakly stable capillary surface immersed in a half-space of R 3 \mathbb{R}^{3} which is minimal or has a contact angle less than or equal to π / 2 \pi/2 must be a half-plane. Using this uniqueness result, we obtain curvature estimates for strongly stable capillary surfaces immersed in a 3-manifold with bounded geometry.

We prove the existence of two smooth families of unbounded domains in ℝ N + 1 {\mathbb{R}^{N+1}} with N ≥ 1 {N\geq 1} such that { - Δ ⁢ u = λ ⁢ u in  ⁢ Ω , u = 0 on  ⁢ ∂ ⁡ Ω , ∂ ν ⁡ u = const on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}{-}\Delta u&\displaystyle=\lambda u&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\\ \displaystyle\partial_{\nu}u&\displaystyle=\mathrm{const}&&\displaystyle% \phantom{}\text{on }\partial\Omega,\end{aligned}\right. admits a sign-changing solution. The domains bifurcate from the straight cylinder B 1 × ℝ {B_{1}\times\mathbb{R}} , where B 1 {B_{1}} is the unit ball in ℝ N {\mathbb{R}^{N}} . These results can be regarded as counterexamples to the Berenstein Conjecture on unbounded domain. Unlike most previous papers in this direction, a very delicate issue here is that there may be two-dimensional kernel space at some bifurcation point. Thus a Crandall–Rabinowitz-type bifurcation theorem from high-dimensional kernel space is also established to achieve the goal.