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Does there exist a simple smooth closed curve $\gamma:[0,2\pi]\to \mathbb C$$\gamma:S^1\to \mathbb C$ such that
$$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
Does there exist a simple smooth closed curve $\gamma:[0,2\pi]\to \mathbb C$ such that
$$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
Does there exist a simple smooth closed curve $\gamma:S^1\to \mathbb C$ such that
$$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
Does there exist a simple smooth closed curve $\gamma:S^1 \to \mathbb C$$\gamma:[0,2\pi]\to \mathbb C$ such that
$$ \int_{S^1} e^{\gamma(t)} \, |\gamma'(t)|\,dt =0?$$$$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
Does there exist a simple smooth closed curve $\gamma:S^1 \to \mathbb C$ such that
$$ \int_{S^1} e^{\gamma(t)} \, |\gamma'(t)|\,dt =0?$$
Does there exist a simple smooth closed curve $\gamma:[0,2\pi]\to \mathbb C$ such that
$$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
