We determine the fundamental limit of the microresonator field uniformity. It can be achieved in a specially designed microresonator, called a bat microresonator, fabricated at the optical fiber surface. We show that the relative nonuniformity of an eigenmode amplitude along the axial length 𝑳𝑳 of an ideal bat microresonator cannot be smaller than 𝟏𝟏 𝟑𝟑 𝝅𝝅 𝟐𝟐 𝒏𝒏 𝒓𝒓 𝟒𝟒 𝝀𝝀 −𝟒𝟒 𝑸𝑸 −𝟐𝟐 𝑳𝑳 𝟒𝟒 , where 𝒏𝒏 𝒓𝒓 , 𝝀𝝀 and 𝑸𝑸 are its refractive index, the eigenmode wavelength and Q-factor. In the absence of losses (𝑸𝑸 = ∞), this eigenmode has the amplitude independent of axial coordinate and zero axial speed (i.e., is stopped) within the length 𝑳𝑳. For a silica microresonator with 𝑸𝑸 = 𝟏𝟏𝟏𝟏 𝟖𝟖 this eigenmode has the axial speed ~ 10 -4 c, where c is the speed of light in vacuum, and its nonuniformity along the length 𝑳𝑳 = 𝟏𝟏𝟏𝟏𝟏𝟏 𝛍𝛍𝛍𝛍 at wavelength 𝝀𝝀 = 𝟏𝟏. 𝟓𝟓 µ𝒎𝒎 is ~ 10 -7 . For a realistic fiber with diameter 100 µm and surface roughness 0.2 nm, the smallest eigenmode nonuniformity is ~ 0.0003. As an application, we consider a bat microresonator evanescently coupled to high Q-factor silica microspheres which serves as a reference supporting the angstrom-precise straight-line translation over the distance 𝑳𝑳 exceeding a hundred microns.