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### Curvature of plane curves on a surface

Let $S$ be a surface and $\gamma$ a curve on $S\subseteq \mathbb{R}^3$ obtained cutting $S$ with a plane. I wuold an upper bound for the curvature of $\gamma$. Are there papers for this topic?
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### Are there some intrinsic invariants of surfaces other than Gaussian curvature?

The principal curvatures of a surface is denoted by $\kappa_{1}, \kappa_{2}$. Let $P(x,y)$ be a polynomial with real coefficients. Assume that $P(\kappa_{1}, \kappa_{2})$ is an intrinsically ...
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### Trefoil Knot Seifert Minimal Surface Equation

I am not very familiar with knot theory nor with minimal surfaces, so I already apologize if my question appears too naive or simple :). I am trying to do the following: Starting from a real ...
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### Surfaces contained in a ball

In this Paper there is a proof that a closed plane curve of length $L$ and curvature bounded by $K$ can be contained inside a circle of radius $L/4 - (\pi - 2)/2K$. Are there similar results for ...
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### Change of length of curve when Fenchel-Nielsen length coordinate increase

Let $F$ be a hyperbolic surface of finite type. Suppose $\alpha$ is a simple closed geodesic and $\beta$ is any closed geodesic intersecting $\alpha$. Consider a Fenchel-Nielsen coordinate of the ...
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### Exact derivation of Von Kármán relation of Gauss curvature

Using relations for surface deformations (in structural mechanics notation) $$u,v,{\epsilon _x, \epsilon_y, \gamma_{xy}}$$ Notations {u,v } have same meaning as displacements in surface theory. ...
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### Families of smooth projective varieties over dvr

Let $R$ be a discrete valuation ring with residue field $k$, an algebraically closed field of characteristic zero and $\pi:X\to \mbox{spec}(R)$ a smooth, projective family of surfaces. Denote by $X_0$...
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### Largest disk inside a spherical domain

It is known (Pestov-Ionin theorem) that if $k_{max}$ is the maximum curvature of a smooth planar loop $\gamma$, then there is a disk of radius $1/k_{max}$ inside $\gamma$. I wonder is there any ...
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### A cubic and six conics problem

I am an electrical engineer system. I live in Viet Nam. I am not a Mathematician. I construct and found a problem as follows: Let a cubic, and five conics $(C_1)$, $(C_2)$, $(C_3)$, $(C_4)$, $(C_5)$. ...
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### Finite union of affinoid is affinoid in proper smooth rigid curves (unless it is everything)

In several papers I have found the surprising statement that finite unions of affinoid subspaces of a proper smooth and connected rigid curve are either the whole curve or again affinoid. Could you ...
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### Umbilic points on Euclidean hypersurfaces

Every smooth embedding of $S^2$ into $\mathbb{R}^3$ has at least one umbilic point (in fact, the recent proof of the Caratheodory conjecture yields two such points). The usual proof of this is to use ...
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### Explicit form of certain polynomials and intersection of curves

Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $C, D$ two effective divisors on $X$ intersecting at finitely many points. Is it true that if $C$ and $D$ intersect in ''low'' number ...
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### When is the product of curves a complete intersection variety

Let $C$ be a smooth, projective curve over $\mathbb{C}$ of genus $g$. Let $L$ be a globally generated line bundle on $C$ and let $h^0(C,L)=r+1$. Let $\phi_L:C\rightarrow\mathbb{P}^r$ be the morphism ...
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### In the $\mathbb{H}^3$ upper half space model, is a hemiellipsoid perpendicular to the plane at infinity a minimal surface?

Given a Jordan curve on the $\mathbb{H}^3$ boundary at infinity, there is a minimal surface (topological disk) in $\mathbb{H}^3$ with the curve as its asymptotic boundary (page.mi.fu-berlin.de/...
Let $X$ be a projective surface over $\mathbb{C}$, let $x\in X$ be the only singular point of $X$. Let $L$ be an ample line bundle on $X$. Consider the blow up $Y$ of $X$ along $x$, $f:Y\... 1answer 118 views ### Self-avoiding/reflecting geodesics on a convex surface Let$S$be the surface of a convex body embedded in$\mathbb{R}^3$. For me$S$is a convex polyhedron, but I am happy to view$S$as a smooth body with positive Gaussian curvature at each point, or ... 1answer 119 views ### On conflicting descriptions for tor of a local cohomology group Let$X$be a smooth projective surface and$C$a Cartier divisor on$X$. Denote by$\mathcal{H}^1_C(\mathcal{O}_X)$the sheaf associated to the presheaf$U \mapsto H^1_{C \cap U}(\mathcal{O}_X|_U)$. ... 1answer 241 views ### What does the action of a 2-torsion line bundle on$Pic^d(C)$do to the number of sections? Let$C$be a smooth projective curve over$\mathbb{C}$. Let$A$be a degree$d$line bundle on$C$, and$M$be a degree 0 line bundle on$C$such that$M^2=\mathcal{O}_C$, that is, it is a 2-torsion ... 1answer 42 views ### Is the length function associated with the twist parameter an increasing function? Let$S$be a closed hyperbolic surface and$x$be an oriented simple closed curve in$S$. Let$y$be an oriented closed curve such that the geometric intersection number between$x$and$y$is ... 0answers 134 views ### Was this particular case of the tube formula known before Weyl and Hotelling? The tube formula is a really nice result in differential geometry which relates the volume of the tubular neighborhood of a submanifold to its intrinsic geometry. It has been proved by Weyl in 1939 ... 0answers 140 views ### Harmonic map heat flow in positive curvature Suppose I wish to relax/smooth a map$\phi:M\rightarrow N$between two surfaces$M,N$embedded in$\mathbb{R}^3$. I could try flowing the map using harmonic heat flow, which (as I understand it) is ... 1answer 996 views ### Why is it so hard to prove Toeplitz' conjecture? I'm a layman in mathematics, so please excuse me in advance for anything in this question that may be inappropriate :D. Well: Four years ago, I was reading (and working to solve the puzzles on) ... 1answer 65 views ### Finding t vlaue in Bezier curve [closed] According to this question, I'm looking for some method to find the t value in Quadratic bezier curve equation: $$B(t)=P_0+t(1-t)P_1+t^2P_2 \space \space where \space 0 ≤ t ≤ 1$$ In this ... 1answer 174 views ### Model over DVR for smooth projective curves Let$C$be a smooth, projective, geometrically irreducible curve of genus at least$2$over a complete discrete valued field$F$of characteristic zero (not necessarily algebraically closed). Let$R$... 2answers 741 views ### Is every closed curve in 3D a geodesic on a genus-0 surface? Let$\gamma$be a smooth, closed, unknotted curve embedded in$\mathbb{R}^3$. Q. Does there always exist a smooth, embedded, genus-zero surface$S \subset \mathbb{R}^3$such that$\gamma$is a ... 0answers 77 views ### Can a cylinder be regarded as a Riemannian manifold? [closed] Consider the surface of a bounded cylinder consisting of a top,bottom and side part together with the metric induced by the euclidean norm on$\mathbb{R}^3$. Can this space be regarded as a Riemannian ... 0answers 103 views ### Kinematics of rolling knots It is well known that there are trefoil knots without tritangent planes, and with 3d printers one can print these beautiful objects and make them roll on planes. (An example:https://www.youtube.com/... 4answers 652 views ### Intrinsic definition of arc length [closed] Is there an intrinsic way of defining the arc length of a curve in$\mathbb{R}^{3}$, that is without resorting to a parametrization of the curve? 1answer 138 views ### Reduction of self-intersections without reducing the geometric intersection Let$F$be a hyperbolic surface. Given a closed curve$a$, let$\bar{a}$denotes the free homotopy class of$a$. Let$i(\bar{a},\bar{b})$denotes the geometric intersection number and$i(\bar{a})$... 1answer 129 views ### Image of any curve can be parametrized without zero derivative? Let$\gamma :[a,b]\to\mathbb{R}^2$be a$C^{1} ([a,b])$injective application. Is it true that there is another continuous parametrization$\rho:[c,d]\to\mathbb{R}^2$such that the following two sets ... 1answer 85 views ### Double coset separability and the existence of vanishing sequences for surface group Definition: Let$G$be a group.$G$is said to be double coset separable if given any finitely generated subgroups$H$and$K$in$G$, given any$g\in G$and$h\not\in HgK$, there exists a finite ... 1answer 315 views ### On equations defining space curves I am reading a text by Prof. Szpiro Tata lectures on equations defining space curves. In the proof of Proposition$1.2$on page$12$he gives explicit description of the defining equations of a local ... 1answer 275 views ### Ivanov's metaconjecture on surface homeomorphisms. In Fifteen problems about MCG Ivanov stated the following metaconjecture: Every object naturally associated to a surface S and having a sufficiently rich structure has$Mod(S)$as its groups of ... 1answer 180 views ### How to find isothermal coordinates equivalent to circles in far limit? I am trying to find the most general rotational coordinate systems for Euclidean 3-space, with the following two defining characteristics: 1) being equivalent to spherical coordinates in the limit of ... 0answers 95 views ### Positive curvature of the boundary away from a point implies regularity? In a paper I'm refereeing, the authors make use of the following geometric fact: Let$U$be an open subset of$\mathbb{R}^2$. If there is a point$p\in \partial U$so that$\partial U \backslash p$... 0answers 229 views ### Hypersurface with singularities I heard once about one open problem. That was about existing a hypersurface of a small degree (5? or 6?) passing through some number (5? 6?) of 3-fold points and 2-fold lines (3 lines?). It was said ... 0answers 70 views ### Topological/numerical constraints for the existence of more than one pencil A famous theorem of Castelnuovo and de Franchis tells us that for$S$a smooth projective complex algebraic surface that for$b \geq 2$, pencils$f : S \to B$of genus$b :=g(B)$are in bijective ... 2answers 99 views ### Planar curves identical to their inverses Is the right strophoid the only planar curve$C$whose inverse curve w.r.t. some circle (in this case: centered on the origin) is identical to$C$? &... 1answer 135 views ### Characterization of$d$-gonal curves on a K3 surface Let$X$be a K3 surface and$C$a curve on$X$. We say that$C$is$d$-gonal if it admits a pencil of degree$d$(and none of smaller degree). I am wondering if there exist characterizations of$d$-... 0answers 210 views ### Are there Zoll pancakes? How flat (flat in pancake-style, not in curvature 0-style), in some extrinsic intuitive measure, can a Zoll surface of revolution (embedded in Euclidean three-space) be? I don't want to impose a ... 0answers 81 views ### Is triple point intersection 'generic' in Teichmuller space? Let$S$be a hyperbolic surface of finite type and$\alpha,\beta$be two closed curves. Consider$X$to be the set of all those points$\chi$in the Teichmuller space$\mathcal{T}(S)$of$S$such that ... 0answers 117 views ### Is there a unique solution? [closed] Let$\mathbf{v}:(a,b)\to\mathbb{R}^2$be a given continuous function and$t_0\in (a,b)$a fixed point. Is it true that the following problem has a unique continuous solution$\mathbf{r}:(a,b)\to\...
Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $||\mathbf{v}(t)||=1,\ \forall t\in (a,b)$. Find all continuous functions $\mathbf{r}:(a,b)\to\mathbb{R}^2$ so that: \$ \...