Min–max theory for free boundary minimal hypersurfaces II: general Morse index bounds and applications

For any smooth Riemannian metric on an $$(n+1)$$ ( n + 1 ) -dimensional compact manifold with boundary $$(M,\partial M)$$ ( M , ∂ M ) where $$3\le (n+1)\le 7$$ 3 ≤ ( n + 1 ) ≤ 7 , we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min–max theory in the Almgren–Pitts setting. We apply our Morse index estimates to prove that for almost every (in the $$C^\infty $$ C ∞ Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M. If $$\partial M$$ ∂ M is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.

Preoperative imaging study in assessment of hazard to Internal Carotid Arteries injury during endoscopic endonasal transsphenoidal surgery

In endoscopic endonasal transsphenoidal procedures, ICA injury occurs up to 3.8%[8]. The highest hazard of injury is in case of contact between the ICA and pituitary gland, during opening of the dura. Preoperative imaging ie. CTA, MRA supports objectively intraoperative techniques of imaging. CTA as well as MRA are essential to access anatomic details in variability of cavernous segment of ICA (C4 ICA). The aim of the study was to measure the space between the pituitary gland and ICA on both sides. Anatomic relations between left and right ICAs were accessed on CTA ( coronal scans) at levels:A- the most concave point of the C4-C5 bend;B- at the most convex point of the C4 bend;C- at the C4 posterior ascending portion. Distances between pituitary gland and ICAs were measured on both sides on MRA (axial scans):A'- at the most concave point of C4-C5 bend;B'- the most convex point of the C4 bend.The Statistica 13 (StatSoft) software was used for the statistical analysis. The Mann Whitney U test was applied to determine differences between the quantitative variables, Spearman's rank correlation coefficients were calculated. The results were considered statistically significant at the level of p<0,05. Distance reduction was shown between pituitary glands and cavernous segment (C4) of ICAs on both sides, which is related to age. This has impact on surgical planning and highlights the risk of ICAs intraoperative injury.

More Turán-Type Theorems for Triangles in Convex Point Sets

We study the following family of problems: Given a set of $n$ points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations. As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact. This leads to 256 extremal Turán-type questions. We give nearly tight (within a $\log n$ factor) bounds for 248 of these questions and show that the remaining 8 questions are all asymptotically equivalent to Stein's longstanding tripod packing problem.