The Einstein-Hilbert action of the space of holomorphic maps from S^2 to CP^k

Let $\mathcal{H}_{n,k}(\Sigma)$ be the space of degree $n\geq 1$ holomorphic maps from a compact Riemann surface $\Sigma$ to $\mathbb{C}P^k$. In the case $\Sigma=S^2$ and $n=1$, the $L^2$ metric on $\mathcal{H}_{1,k}(S^2)$ was computed exactly by Speight. In this paper, the Ricci curvature tensor and the scalar curvature on $\mathcal{H}_{1,k}(S^2)$ are determined explicitly for $k\geq 2$. An exact direct computation of the Einstein-Hilbert action with respect to the $L^2$ metric on $\mathcal{H}_{1,k}(S^2)$ is made and shown to coincide with a formula conjectured by Baptista.Comment: 12 pages - updated versio

Quiet a Puzzle: A Case Report of 15 year Old Female with Severe Anemia Due To Scurvy

Scurvy is historically thought of a disease affecting sailors in the seventeenth century. Vitamin C deficiency is rare in the developed world. It is mainly found in patients with poor nutrition and specifically those with developmental delay. The spectrum of scurvy is varied and includes dermatological, dental, bone and systemic manifestations. We present to you a case of a 15 year old female with a history of trisomy 21 presenting with severe anemia as a direct cause of vitamin C deficiency. She presented with a one month history of progressive bilateral lower extremity bruising and inability to walk. Physical exam notable for trisomy 21 features. Bilateral swelling in lower extremities with the right calf more swollen than left. Perifollicular hyperkeratotic papules with surrounding pinpoint hemorrhage in upper extremities with coiled hair. Lower extremity had large areas of ecchymosis associated with significant tenderness. Reminder of the physical exam and review of the systems is unremarkable. Consent for photographs of exam findings were obtained from parents. She underwent extensive work up which showed normocytic anemia hgb 8.7 g/dL. Elevated D dimer. ANA +. Normal Iron studies, Comprehensive metabolic panel, coagulation studies. Autoimmune work was unremarkable. Hemolysis labs were grossly unremarkable. Hematological work up including ADAMTS12, Paroxysmal nocturnal hemoglobinuria , Factor deficiencies, platelet function assay, protein C , S were unremarkable. Duplex ultrasound , CT and CTA of lower extremities were unremarkable. Vitamin C levels were low and barely detectable at \u3c0.1mg/dl. Patient was started on Vitamin C supplementation of 250 mg twice a day and iron supplementation. She was found to have low B12 and vitamin D.. Significant improvement was notable in Three weeks of treatment as the patient was able to walk. In addition to near complete resolution of her ecchymosis and improvement in her anemia