Questions tagged [curves-and-surfaces]

A surface is a generalization of a plane which needs not be flat, that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line

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Are isotopic transversal curves on a foliated surface transversally isotopic?

Let $F$ be an orientable surface (possibly with boundary) with a foliation $\cal F$ with $k$-prong saddle singularities only for $k\geq 3,$ (as in figure borrowed from Farb-Margalit book). Suppose ...
Adam's user avatar
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7 votes
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Which surfaces embedded in $\mathbb{R}^3$ have only axially-symmetric sections?

Dmitry Ryabogin and I considered the following question some time ago, but got nowehere: Let $M$ be a (smooth or algebraic) surface in $\mathbb{R}^3$. Suppose that for every section $S$ (an ...
Michael's user avatar
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9 votes
1 answer
488 views

Degeneration of curves inside a family of surfaces

We are interested in understanding how a higher genus curve in a smooth surface of general type can degenerate into the non-normal locus of the limit of a degeneration of surfaces. More precisely: ...
Srks's user avatar
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3 votes
1 answer
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Symmetry of functions on $S^2$

Let $f$ be a continuous function on $S^2$ and suppose there exists a constant $C>0$ such that for every $\mathcal{R} \in SO(3)$ the area of every connected component of $\{f(x)\geq f(\mathcal{R}x)\}...
Matchmaticians's user avatar
1 vote
0 answers
288 views

Connected component of Picard group of a genus $2$ curve and its Jacobian over an imperfect field

Let $H/k$ be a genus $2$ curve. Consider the function field of $H$ given by $k(H)$. Take the base extension of $H$ to $k(H)$, namely $\hat H:=H\otimes_k k(H)$. Consider the Jacobian $J$ of the curve $...
Eduardo R. Duarte's user avatar
7 votes
1 answer
156 views

Estimate of number of boundary components of a compact Riemannian 2-surface

Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $-1$ and the diameter is at most $D$. Assume that near the boundary the ...
asv's user avatar
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7 votes
1 answer
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Estimate of area of 2-dimensional surface

Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $\kappa$, the diameter is at most $D$, and the second fundamental form ...
asv's user avatar
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5 votes
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Relation between point pushing pseudo-Anosov map and the minimum length

Let $S$ be a closed hyperbolic surface. Suppose $Mod(S)$ denotes the mapping class groups and $T(F)$ denotes the Teichmüller space. By Birman exact sequence we get the point pushing map $Push:\pi_1(S,...
Cusp's user avatar
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1 vote
1 answer
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Splines with bounded first derivative?

I have a set of points $(x_i,y_i)\in{\mathbb R}_+\times{\mathbb R}$, $i=1,...,n$, ($x_i$ are the independent variables and $y_i$ are the dependent variables or responses) that I want to fit using ...
Fito's user avatar
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2 votes
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Regular surfaces with boundary and $C^1$ domains

I would like to ask about the equivalence between these two definitions for a $C^1$ domain. In the book Vector Analysis Versus Vector Calculus, we have: Definition 8.2.1: Let $\mathbb{H}^k=\{(t_1,\...
user39756's user avatar
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5 votes
2 answers
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homogeneous surface in $\mathbb{R}^4$

It is well known that the only homogeneous surfaces in $\mathbb{R}^3$ are the spheres, cylinders or planes. My question is about other examples in dimension $4$. Such a surface should have "constant ...
Paul's user avatar
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7 votes
1 answer
326 views

Uniformisation for non simple closed curves

Given a simple closed curve in the plane, there is a homeomorphism from the unit open disk to the interior of the curve. The homeomorphism can be taken conformal, this is the uniformisation theorem. ...
coudy's user avatar
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7 votes
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Can generalization of a generalization Pascal theorem, Pappus theorem to Higher Dimensions? [closed]

Please see a chain of six circles associated with a conic. This is a generalization of Pascal theorem, Pappus theorem. I reformulate as following: Let $1, 2, 3, 4, 5, 6$ be six arbitrary points in a ...
Cố Gắng Lên's user avatar
7 votes
3 answers
427 views

Identity involving an improper integral (with geometric application)

Is it (for some reason) true that $\lim_{c\to 0^+}\int_c^{\pi/2}\frac{c}{t}\sqrt\frac{1+t^2}{t^2-c^2}dt=\frac{\pi}{2}$? Numerical evidence (from Mathematica): when $c=1/5$, the integral is $\...
macbeth's user avatar
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1 vote
1 answer
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Smoothness Conditions for Planar "Mock-parametric" Spline Interpolation

By "mock-parametric" interpolating curves I understand a class of curves that connect a discrete sequence of points with a predefined degree of smoothness and, that correspond to a non-...
Manfred Weis's user avatar
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4 votes
1 answer
131 views

Name for Curves from Driving on Smooth Manifolds

Is there already name for the generalization of Clothoids to curves on smooth manifolds, i.e. where the curve's curvature depends linearly on the curve's length-parameter? In the euclidean plane ...
Manfred Weis's user avatar
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2 votes
0 answers
135 views

A gradient trajectory connecting boundary components in an annulus

In a course of writing a paper I realised that I need a lemma below, and found a half page proof. I have not seen this lemma before and wonder if maybe someone here knows this lemma or can prove it in ...
aglearner's user avatar
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4 votes
1 answer
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Convex hull of a connected subset on a complete surface of non-positive curvature

Let $S$ be a simply connected surface, possibly with boundary components, with a smooth complete metric of non-positive curvature. Let $X\subset S$ be a closed connected subset. I would like to know ...
aglearner's user avatar
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1 vote
1 answer
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Polar coordinates of a set with different radius and angle

Let $M$ be a $2$-dimensional Riemannian manifold and let $U\subset M$ be an open set. Suppose there exist polar coordinates $(r,\theta)$ with center $q\in M$ such that $$U=\lbrace{ (r,\theta): 0<...
Sammyy Delbrin's user avatar
2 votes
1 answer
655 views

Castelnuovo and Artin contractibility criteria for families

In the course of a proof, I need some version of Castelnuovo and Artin's contractibility criteria for a family of surfaces. Say I have a family (flat is probably needed, in order to compare ...
Stefano's user avatar
  • 625
2 votes
1 answer
279 views

Regularity of the reparametrization map between curves [closed]

I am looking for a reference for the following kind of results. Let $\Gamma$ be the space of Lipschitz curves $\text{Lip}([0,1]; \mathbb R^d)$ equipped with the sup norm. Let $B$ be a Borel subset of ...
Romeo's user avatar
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7 votes
2 answers
346 views

Most general version for the Gauss-Bonnet theorem for polygons

Suppose $M$ is a 2-dimensional smooth Riemannian manifold and $P\subset M$ is an open and connected subset with compact closure and a piecewise geodesic boundary. My question is: What further ...
Sammyy Delbrin's user avatar
2 votes
1 answer
116 views

Construction of a linear Weingarten surface from a space curve

In Ivey and Landsberg's book Cartan for Beginners, the end paragraph of example 5.8.2 claims that linear Weingarten surfaces can be constructed by a space curve. They cite an older book from 1945 that ...
TK-421's user avatar
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2 votes
0 answers
127 views

intersections between closed curves on surfaces

I would like to find a result telling me that two simple closed curves $\alpha$ and $\beta$ (on a non-orientable surface $S$) are in minimal position if and only if there is not a disk in $S$ whose ...
user104820's user avatar
17 votes
2 answers
1k views

A variant of the Monge-Cayley-Salmon theorem?

Suppose one has a smooth non-degenerate curve $\gamma: [0,1] \to {\bf R}^n$ into Euclidean space (thus $\gamma'$ never vanishes), with the property that the velocity $\gamma'(t)$ and acceleration $\...
Terry Tao's user avatar
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4 votes
1 answer
212 views

Finding the shortest curve that is at distance $\epsilon$ of every point of a surface

Let $M$ be a compact connected (smooth) surface (possibly with boundary) in $\mathbb{R}^3$ and $\epsilon>0$ a constant. Is there (and if there's not, what conditions on ($M$, $\epsilon$) should ...
LCO's user avatar
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2 votes
1 answer
83 views

"Slope Analogue" of Clothoids

It is well known, that the characterizing property of Clothoids is, that their curvature is proportional to length; that is also the reason, why they are used as design elements e.g. in road design. ...
Manfred Weis's user avatar
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5 votes
0 answers
433 views

Examples of spiraling geodesics?

Does there exist a closed, bounded surface $S$ embedded in $\mathbb{R}^3$ that has a geodesic $\gamma$ that spirals around a point $x$, getting closer and closer, but never reaching $x$? Here I ...
Joseph O'Rourke's user avatar
26 votes
1 answer
820 views

Disc bounded by a plane curve

Let $\Sigma$ be a sphere topologically embedded into $\mathbb{R}^3$. Is it always possible to find a disc $\Delta\subset\Sigma$ which is bounded by a plane curve? It is easy to find an open disc ...
Anton Petrunin's user avatar
2 votes
0 answers
126 views

Is there an analog of Reidemeister's theorem for braids in a surface?

Reidemeister's classical theorem describes the set of links in $\mathbb R^3$ up to isotopy as the set $\{ \textrm{diagrams in } \mathbb R^2\textrm{ with crossings}\}$ modulo certain local relations on ...
Peter Samuelson's user avatar
5 votes
0 answers
377 views

Gage-Grayson-Hamilton curve-shortening flow, at an angle

The Gage-Grayson-Hamilton curve-shortening flows along the normal to the curve:                     &...
Joseph O'Rourke's user avatar
2 votes
0 answers
165 views

Moving curves in minimal position to geodesics on hyperbolic surfaces

Let $\Sigma$ be a hyperbolic surface possibly with non-empty boundary and punctures. A closed curve $\gamma$ if filling, if any simple closed curve intersects any representative of free homotopy ...
Michał Marcinkowski's user avatar
2 votes
0 answers
198 views

Universal chord theorem for curves

Let $\mathrm{\gamma} : [0,1] \to \mathbb{R}^2$ be a piecewise smooth, simple plane curve. Assume $\gamma(0) = (0,0)$, $\gamma(1) = (1,0)$ and that the slope of the tangent is not $0$ wherever it's ...
user95393's user avatar
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3 votes
0 answers
126 views

Behaviour of geodesics on surfaces as one of the two endpoints moves slightly

Let $u$ and $v$ be two points on a surface (I guess, a Riemann surface) $\Sigma$ such that there is a unique geodesic between $u$ and $v$ on $\Sigma$. Now let $l$ be an arbitrary line that passes ...
Hooman's user avatar
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3 votes
0 answers
308 views

Does a rectangle exist on any Jordan curve?

Let $C$ be a Jordan curve in $\mathbb{R}^2$. Does there exist points $P,Q,R,S$ on $C$ such that quadrangle $PQRS$ is a non-degenerate rectangle?
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2 votes
2 answers
181 views

a continuous version of axiom of choice?

Let $L_t(s):S^1 \rightarrow \mathbb{R}^2(t\in [0,1])$ be a Jordan curve, $O(t)$ be its interior and $H(t,s)=L_t(s)$. If $H$ is a homotopy from $L_0(s)$ to$L_1(s)$, does there exist a continuous ...
user avatar
0 votes
1 answer
97 views

Isometric and conformally equivalent surfaces in $\mathbb R^d$ with the same boundary

Let $X$ be a compact smooth surface in $\mathbb R^d$ with smooth boundary $\partial X$. Is it possible to find another smooth surface $Y$ in $\mathbb R^d$ such that $\partial X = \partial Y$ and $Y$ ...
Appliqué's user avatar
  • 1,269
2 votes
0 answers
162 views

Convexity of length function for surfaces with boundary

In the paper "The Nielsen realization problem" (here), Kerckhoff proved that the length function on the Teichmüller for closed surface is convex. In his paper "Geodesic length functions and the ...
Cusp's user avatar
  • 1,703
2 votes
0 answers
255 views

How to descend a line bundle from the normalization of a surface?

Let $\tilde S \overset{\nu}{\to} S$ be the normalization of a projective surface $S$ over a field $k$. Assume for now that $S$ is obtained from $\tilde S$ by gluing together two disjoint curves $C_1$ ...
Emre's user avatar
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4 votes
0 answers
641 views

Research topics in Curves and Surfaces [closed]

I advance that I'm not a mathematician but I'm an undergraduate student of mathematics. In my courses at university I have studied a bit of Differential Geometry, in particoular differential geometry ...
Vincenzo Zaccaro's user avatar
1 vote
1 answer
205 views

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?
Vincenzo Zaccaro's user avatar
17 votes
2 answers
1k views

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 ...
Ali Taghavi's user avatar
2 votes
0 answers
450 views

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 ...
Aobara's user avatar
  • 181
8 votes
2 answers
365 views

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 ...
Vincenzo Zaccaro's user avatar
1 vote
0 answers
141 views

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 ...
Cusp's user avatar
  • 1,703
1 vote
0 answers
188 views

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. ...
Narasimham's user avatar
1 vote
0 answers
63 views

decomposition of codim 1 currents

Let $T$ be an codimension one rectifiable mass-minimizing current. If $\partial T\llcorner B_R(a)=0$, then it is possible to write $T$ as an infinite sum of oriented boundaries: $T\llcorner B_R(a)=\...
leander's user avatar
  • 43
1 vote
0 answers
96 views

A third degree surface and a touching sphere [closed]

Let consider a surface $z=1/(xy)$ and a sphere defined by $(x-1.5)^2+(y-1.5)^2+(z-1.5)^2=3/4$. The sphere touches the surface at (1,1,1). Is it possible to prove that point (1,1,1) is the only ...
Sergei's user avatar
  • 1,540
4 votes
1 answer
617 views

Polars of algebraic curves and surfaces

I asked this on Math.StackExchange, but received no response, so trying here ... A paper I'm reading says the following ... With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let $F(\mathbf{x})...
bubba's user avatar
  • 629
2 votes
1 answer
444 views

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 $X_0$...
user43198's user avatar
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