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We prove that optimal lower eigenvalue estimates of Zhong-Yang type as well as a Cheng-type upper bound for the first eigenvalue hold on closed manifolds assuming only a Kato condition on the negative part of the Ricci curvature.
This generalizes all earlier results on Lp-curvature assumptions.
Moreover, we introduce the Kato condition on compact manifolds with boundary with respect to the Neumann Laplacian, leading to Harnack estimates for the Neumann heat kernel and lower bounds for all Neumann eigenvalues, which provides a first insight in handling variable Ricci curvature assumptions in this case.
Review exploring the regulation of PHYTOCHROME-INTERACTING FACTORS by light, their role in abiotic stress tolerance and plant architecture, and their influence on crop productivity.
Light is a key determinant for plant growth, development, and ultimately yield. Phytochromes, red/far-red photoreceptors, play an important role in plant architecture, stress tolerance, and productivity. In the model plant Arabidopsis, it has been shown that PHYTOCHROME-INTERACTING FACTORS (PIFs; bHLH transcription factors) act as central hubs in the integration of external stimuli to regulate plant development. Recent studies have unveiled the importance of PIFs in crops. They are involved in the modulation of plant architecture and productivity through the regulation of cell division and elongation in response to different environmental cues. These studies show that different PIFs have overlapping but also distinct functions in the regulation of plant growth. Therefore, understanding the molecular mechanisms by which PIFs regulate plant development is crucial to improve crop productivity under both optimal and adverse environmental conditions. In this review, we discuss current knowledge of PIFs acting as integrators of light and other signals in different crops, with particular focus on the role of PIFs in responding to different environmental conditions and how this can be used to improve crop productivity.
Plants have evolved numerous molecular strategies to cope with perturbations in environmental temperature, and to adjust growth and physiology to limit the negative effects of extreme temperature. One of the strategies involves alternative splicing of primary transcripts to encode alternative protein products or transcript variants destined for degradation by nonsense-mediated decay. Here, we review how changes in environmental temperature-cold, heat, and moderate alterations in temperature-affect alternative splicing in plants, including crops. We present examples of the mode of action of various temperature-induced splice variants and discuss how these alternative splicing events enable favourable plant responses to altered temperatures. Finally, we point out unanswered questions that should be addressed to fully utilize the endogenous mechanisms in plants to adjust their growth to environmental temperature. We also indicate how this knowledge might be used to enhance crop productivity in the future.