Min-max theory for GG-invariant minimal hypersurfaces

In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$, and a compact Lie group $G$ acting as isometries on $M$ with cohomogeneity at least $3$. After adapting the Almgren-Pitts min-max theory to a $G$-equivariant version, we show the existence of a nontrivial closed smooth embedded $G$-invariant minimal hypersurface $\Sigma\subset M$ provided that the union of non-principal orbits forms a smooth embedded submanifold of $M$ with dimension at most $n-2$. Moreover, we also build upper bounds as well as lower bounds of $(G,p)$-width which are analogs of the classical conclusions derived by Gromov and Guth. An application of our results combined with the work of Marques-Neves shows the existence of infinitely many $G$-invariant minimal hypersurfaces when ${\rm Ric}_M>0$ and orbits satisfy the same assumption above

Min-max theory for free boundary G-invariant minimal hypersurfaces

Given a compact Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$ and $\partial M\neq\emptyset$, the free boundary min-max theory built by Martin Man-Chun Li and Xin Zhou shows the existence of a smooth almost properly embedded minimal hypersurface with free boundary in $\partial M$. In this paper, we generalize their constructions into equivariant settings. Specifically, let $G$ be a compact Lie group acting as isometries on $M$ with cohomogeneity at least $3$. Then we show that there exists a nontrivial smooth almost properly embedded $G$-invariant minimal hypersurface with free boundary. Moreover, if the Ricci curvature of $M$ is non-negative and $\partial M$ is strictly convex, then there exist infinitely many properly embedded $G$-invariant minimal hypersurfaces with free boundary.Comment: Accepted by Advances in Mathematic

Equivariant Morse index of min-max GG-invariant minimal hypersurfaces

For a closed Riemannian manifold $M^{n+1}$ with a compact Lie group $G$ acting as isometries, the equivariant min-max theory gives the existence and the potential abundance of minimal $G$-invariant hypersurfaces provided $3\leq {\rm codim}(G\cdot p) \leq 7$ for all $p\in M$. In this paper, we show a compactness theorem for these min-max minimal $G$-hypersurfaces and construct a $G$-invariant Jacobi field on the limit. Combining with an equivariant bumpy metrics theorem, we obtain a $C^\infty_G$-generic finiteness result for min-max $G$-hypersurfaces with area uniformly bounded. As a main application, we further generalize the Morse index estimates for min-max minimal hypersurfaces to the equivariant setting. Namely, the closed $G$-invariant minimal hypersurface $\Sigma\subset M$ constructed by the equivariant min-max on a $k$-dimensional homotopy class can be chosen to satisfy ${\rm Index}_G(\Sigma)\leq k$.Comment: Final version. Accepted by Mathematische Annale

Improved Hebey-Vaugon conjecture on equivariant Yamabe invariants in dimension 3

Consider a closed connected $3$-manifold $M$ acted diffeomorphically on by a compact Lie group $G$ with at least one orbit of finite cardinality. We show an upper bound for the $G$-equivariant Yamabe invariant $\sigma_G(M)$ under certain topological assumptions, which improved a conjecture of Hebey-Vaugon.Comment: 10 pages, 2 figures, comments are welcome

Min-max theory for free boundary minimal hypersurfaces in locally wedge-shaped manifolds

We develop a min-max theory for the area functional in the class of locally wedge-shaped manifolds. Roughly speaking, a locally wedge-shaped manifold is a Riemannian manifold that is allowed to have both boundary and certain types of edges. Fix a dimension $3 \le n+1 \le 6$. As our main theorem, we prove that every compact locally wedge-shaped manifold $M^{n+1}$ with acute wedge angles contains a locally wedge-shaped free boundary minimal hypersurface $\Sigma^n$ which is smooth in its interior and on its faces and is $C^{2,\alpha}$ up to and including its edge. We can also handle the case of 90 degree wedge angles under an additional assumption.Comment: 47 pages, 6 figures, comments are welcome

Generalized S1S^1-stability theorem

We use the equivariant $\mu$-bubbles technique to prove that for any compact manifold $M^n$ with non-empty boundary, $n\in\{3,5,6\}$, the Yamabe invariant of $M^n$ is positive if and only if the Yamabe invariant of $M^n\times S^1$ is positive. This generalized the $S^1$-stability conjecture of Rosenberg to compact manifolds with boundary.Comment: 14 pages, comments are welcome

Curvature estimates for stable free boundary minimal hypersurfaces in locally wedge-shaped manifolds

In this paper, we consider locally wedge-shaped manifolds, which are Riemannian manifolds that are allowed to have both boundary and certain types of edges. We define and study the properties of free boundary minimal hypersurfaces inside locally wedge-shaped manifolds. In particular, we show a compactness theorem for free boundary minimal hypersurfaces with curvature and area bounds in a locally wedge-shaped manifold. Additionally, using Schoen-Simon-Yau's estimates, we also prove a Bernstein-type theorem indicating that, under certain conditions, a stable free boundary minimal hypersurface inside a Euclidean wedge must be a portion of a hyperplane. As our main application, we establish a curvature estimate for sufficiently regular free boundary minimal hypersurfaces in a locally wedge-shaped manifold. We expect this curvature estimate will be useful for establishing a min-max theory for the area functional in wedge-shaped spaces.Comment: 28 pages, 6 figures, comments are welcome

Stabilization of Fast Pyrolysis Liquids from Biomass by Mild Catalytic Hydrotreatment:Model Compound Study

Repolymerization is a huge problem in the storage and processing of biomass pyrolysis liquid (PL). Herein, to solve the problem of repolymerization, mild catalytic hydrotreatment of PL was conducted to convert unstable PL model compounds (hydroxyacetone, furfural, and phenol) into stable alcohols. An Ni/SiO2 catalyst was synthesized by the deposition-precipitation method and used in a mild hydrotreatment process. The mild hydrotreatment of the single model compound was studied to determine the reaction pathways, which provided guidance for improving the selectivity of stable intermediate alcohols through the control of reaction conditions. More importantly, the mild hydrotreatment of mixed model compounds was evaluated to simulate the PL more factually. In addition, the effect of the interaction between hydroxyacetone, furfural, and phenol during the catalytic hydrotreatment was also explored. There was a strange phenomenon observed in that phenol was not converted in the initial stage of the hydrotreatment of mixed model compounds. Thermogravimetric analysis (TGA), Ultraviolet-Raman (UV-Raman), and Brunauer−Emmett−Teller (BET) characterization of catalysts used in the hydrotreatment of single and mixed model compounds demonstrated that this phenomenon did not mainly arise from the irreversible deactivation of catalysts caused by carbon deposition, but the competitive adsorption among hydroxyacetone, furfural, and phenol during the mild hydrotreatment of mixed model compounds

Effects of different fertilization conditions and different geographical locations on the diversity and composition of the rhizosphere microbiota of Qingke (Hordeum vulgare L.) plants in different growth stages

IntroductionThe excessive use of chemical fertilizer causes increasing environmental and food security crisis. Organic fertilizer improves physical and biological activities of soil. Rhizosphere microbiota, which consist of highly diverse microorganisms, play an important role in soil quality. However, there is limited information about the effects of different fertilization conditions on the growth of Qingke plants and composition of the rhizosphere microbiota of the plants.MethodsIn this study, we characterized the rhizosphere microbiota of Qingke plants grown in three main Qingke-producing areas (Tibet, Qinghai, and Gansu). In each of the three areas, seven different fertilization conditions (m1–m7, m1: Unfertilized; m2: Farmer Practice; m3: 75% Farmer Practice; m4: 75% Farmer Practice +25% Organic manure; m5: 50% Farmer Practice; m6: 50% Farmer Practice +50% Organic manure; m7: 100% Organic manure) were applied. The growth and yields of the Qingke plants were also compared under the seven fertilization conditions.ResultsThere were significant differences in alpha diversity indices among the three areas. In each area, differences in fertilization conditions and differences in the growth stages of Qingke plants resulted in differences in the beta diversity of the rhizosphere microbiota. Meanwhile, in each area, fertilization conditions, soil depths, and the growth stages of Qingke plants significantly affected the relative abundance of the top 10 phyla and the top 20 bacterial genera. For most of microbial pairs established through network analysis, the significance of their correlations in each of the microbial co-occurrence networks of the three experimental sites was different. Moreover, in each of the three networks, there were significant differences in relative abundance and genera among most nodes (i.e., the genera Pseudonocardia, Skermanella, Pseudonocardia, Skermanella, Aridibacter, and Illumatobacter). The soil chemical properties (i.e., TN, TP, SOM, AN, AK, CEC, Ca, and K) were positively or negatively correlated with the relative abundance of the top 30 genera derived from the three main Qingke-producing areas (p < 0.05). Fertilization conditions markedly influenced the height of a Qingke plant, the number of spikes in a Qingke plant, the number of kernels in a spike, and the fresh weight of a Qingke plant. Considering the yield, the most effective fertilization conditions for Qingke is combining application 50% chemical fertilizer and 50% organic manure.ConclusionThe results of the present study can provide theoretical basis for practice of reducing the use of chemical fertilizer in agriculture

Rapid screening of Salmonella enterica serovars Enteritidis, Hadar, Heidelberg and Typhimurium using a serologically-correlative allelotyping PCR targeting the O and H antigen alleles

<p>Abstract</p> <p>Background</p> <p>Classical <it>Salmonella </it>serotyping is an expensive and time consuming process that requires implementing a battery of O and H antisera to detect 2,541 different <it>Salmonella enterica </it>serovars. For these reasons, we developed a rapid multiplex polymerase chain reaction (PCR)-based typing scheme to screen for the prevalent <it>S. enterica </it>serovars Enteritidis, Hadar, Heidelberg, and Typhimurium.</p> <p>Results</p> <p>By analyzing the nucleotide sequences of the genes for O-antigen biosynthesis including <it>wb</it>a operon and the central variable regions of the H1 and H2 flagellin genes in <it>Salmonella</it>, designated PCR primers for four multiplex PCR reactions were used to detect and differentiate <it>Salmonella </it>serogroups A/D1, B, C1, C2, or E1; H1 antigen types i, g, m, r or z₁₀; and H2 antigen complexes, I: 1,2; 1,5; 1,6; 1,7 or II: e,n,x; e,n,z₁₅. Through the detection of these antigen gene allele combinations, we were able to distinguish among <it>S. enterica </it>serovars Enteritidis, Hadar, Heidelberg, and Typhimurium. The assays were useful in identifying <it>Salmonella </it>with O and H antigen gene alleles representing 43 distinct serovars. While the H2 multiplex could discriminate between unrelated H2 antigens, the PCR could not discern differences within the antigen complexes, 1,2; 1,5; 1,6; 1,7 or e,n,x; e,n,z₁₅, requiring a final confirmatory PCR test in the final serovar reporting of <it>S. enterica</it>.</p> <p>Conclusion</p> <p>Multiplex PCR assays for detecting specific O and H antigen gene alleles can be a rapid and cost-effective alternative approach to classical serotyping for presumptive identification of <it>S. enterica </it>serovars Enteritidis, Hadar, Heidelberg, and Typhimurium.</p